# Generating Flows - Continuously Iterated Generating Functions

See Schroeder Summations - Experimental Results for notes on limitations and implementation. Uses the Iterate.m package.

### The Continuous Iteration the Generating Functions of Parabolic Rational Neutral Maps with a Fixed Point at Zero

Let be an the exponential generating function of a combinatorial structure where and , then is a parabolic rational neutral map with a fixed point at zero. The flow of is the exponential generating function of a family of combinatorial structures polynomial in and .

`{0, 1, n*a[2], (n*(3*(-1 + n)*a[2]^2 + 2*a[3]))/2,  (n*(3*(3 - 5*n + 2*n^2)*a[2]^3 + 10*(-1 + n)*a[2]*a[3] + 2*a[4]))/2,  (n*(15*(-8 + 18*n - 13*n^2 + 3*n^3)*a[2]^4 + 5*(37 - 63*n + 26*n^2)*a[2]^2*a[3] +     45*(-1 + n)*a[2]*a[4] + 6*(5*(-1 + n)*a[3]^2 + a[5])))/6,  (n*(45*(62 - 175*n + 178*n^2 - 77*n^3 + 12*n^4)*a[2]^5 +     30*(-184 + 427*n - 320*n^2 + 77*n^3)*a[2]^3*a[3] + 360*(4 - 7*n + 3*n^2)*a[2]^2*     a[4] + 28*(-1 + n)*a[2]*(10*(-7 + 5*n)*a[3]^2 + 9*a[5]) +     12*(35*(-1 + n)*a[3]*a[4] + 2*a[6])))/24,  (n*(63*(-314 + 1075*n - 1405*n^2 + 875*n^3 - 261*n^4 + 30*n^5)*a[2]^6 +     42*(1144 - 3350*n + 3545*n^2 - 1600*n^3 + 261*n^4)*a[2]^4*a[3] +     420*(-31 + 75*n - 59*n^2 + 15*n^3)*a[2]^3*a[4] + 28*(-1 + n)*a[2]^2*     (5*(193 - 262*n + 85*n^2)*a[3]^2 + 18*(-5 + 4*n)*a[5]) +     56*(-1 + n)*a[2]*(5*(-31 + 23*n)*a[3]*a[4] + 6*a[6]) +     4*(70*(7 - 12*n + 5*n^2)*a[3]^3 + 168*(-1 + n)*a[3]*a[5] +       3*(35*(-1 + n)*a[4]^2 + 2*a[7]))))/24,  (n*(63*(1298 - 5301*n + 8510*n^2 - 6900*n^3 + 3002*n^4 - 669*n^5 + 60*n^6)*a[2]^7 +     21*(-11282 + 40257*n - 54830*n^2 + 35625*n^3 - 11108*n^4 + 1338*n^5)*a[2]^5*a[3] +     105*(628 - 1935*n + 2162*n^2 - 1035*n^3 + 180*n^4)*a[2]^4*a[4] +     28*(-1 + n)*a[2]^3*(4*(-1652 + 3298*n - 2062*n^2 + 413*n^3)*a[3]^2 +       15*(32 - 49*n + 18*n^2)*a[5]) + 14*(-1 + n)*a[2]^2*     (75*(68 - 97*n + 33*n^2)*a[3]*a[4] + 2*(-73 + 62*n)*a[6]) +     2*(-1 + n)*a[2]*(210*(78 - 107*n + 35*n^2)*a[3]^3 + 35*(-77 + 58*n)*a[4]^2 +       56*(-76 + 59*n)*a[3]*a[5] + 108*a[7]) +     4*(70*(26 - 45*n + 19*n^2)*a[3]^2*a[4] + 126*(-1 + n)*a[3]*a[6] +       3*(63*(-1 + n)*a[4]*a[5] + a[8]))))/12}`
`{0, 1, n*a[2], (n*(12*(-1 + n)*a[2]^2 + 12*a[3]))/12,  (n*(24*(3 - 5*n + 2*n^2)*a[2]^3 + 120*(-1 + n)*a[2]*a[3] + 48*a[4]))/48,  (n*(240*(-8 + 18*n - 13*n^2 + 3*n^3)*a[2]^4 + 120*(37 - 63*n + 26*n^2)*a[2]^2*a[3] +     2160*(-1 + n)*a[2]*a[4] + 6*(180*(-1 + n)*a[3]^2 + 120*a[5])))/720,  (n*(1440*(62 - 175*n + 178*n^2 - 77*n^3 + 12*n^4)*a[2]^5 +     1440*(-184 + 427*n - 320*n^2 + 77*n^3)*a[2]^3*a[3] +     34560*(4 - 7*n + 3*n^2)*a[2]^2*a[4] + 56*(-1 + n)*a[2]*     (360*(-7 + 5*n)*a[3]^2 + 1080*a[5]) + 12*(5040*(-1 + n)*a[3]*a[4] + 1440*a[6])))/  17280, (n*(4032*(-314 + 1075*n - 1405*n^2 + 875*n^3 - 261*n^4 + 30*n^5)*a[2]^6 +     4032*(1144 - 3350*n + 3545*n^2 - 1600*n^3 + 261*n^4)*a[2]^4*a[3] +     80640*(-31 + 75*n - 59*n^2 + 15*n^3)*a[2]^3*a[4] +     112*(-1 + n)*a[2]^2*(180*(193 - 262*n + 85*n^2)*a[3]^2 + 2160*(-5 + 4*n)*a[5]) +     112*(-1 + n)*a[2]*(720*(-31 + 23*n)*a[3]*a[4] + 4320*a[6]) +     4*(15120*(7 - 12*n + 5*n^2)*a[3]^3 + 120960*(-1 + n)*a[3]*a[5] +       3*(20160*(-1 + n)*a[4]^2 + 10080*a[7]))))/120960,  (n*(8064*(1298 - 5301*n + 8510*n^2 - 6900*n^3 + 3002*n^4 - 669*n^5 + 60*n^6)*a[2]^7 +     4032*(-11282 + 40257*n - 54830*n^2 + 35625*n^3 - 11108*n^4 + 1338*n^5)*a[2]^5*     a[3] + 40320*(628 - 1935*n + 2162*n^2 - 1035*n^3 + 180*n^4)*a[2]^4*a[4] +     224*(-1 + n)*a[2]^3*(144*(-1652 + 3298*n - 2062*n^2 + 413*n^3)*a[3]^2 +       1800*(32 - 49*n + 18*n^2)*a[5]) + 56*(-1 + n)*a[2]^2*     (10800*(68 - 97*n + 33*n^2)*a[3]*a[4] + 1440*(-73 + 62*n)*a[6]) +     4*(-1 + n)*a[2]*(45360*(78 - 107*n + 35*n^2)*a[3]^3 + 20160*(-77 + 58*n)*a[4]^2 +       40320*(-76 + 59*n)*a[3]*a[5] + 544320*a[7]) +     4*(60480*(26 - 45*n + 19*n^2)*a[3]^2*a[4] + 544320*(-1 + n)*a[3]*a[6] +       3*(181440*(-1 + n)*a[4]*a[5] + 40320*a[8]))))/483840}`

### Hierarchies of Height n

{1,1/2,1/8,0,1/32,-7/128,1/128,159/256}      1/2 A052122
{0,1,1,1,1,1,1,1,1}                                           1
{1,2,5,15,52,203,877,4140}                            2 A000110
{1,3,12,60,358,2471,19302,167894}              3 A000258
{1,4,22,154,1304,12915,146115,1855570}    4 A000307
{1,-1,2,-6,24,-120,720,-5040}                      -1 A000142
{1,-2,7,-35,228,-1834,17582,-195866}        -2 A003713

### Fibonacci numbers

{1,1,2,3,5,8,13,21}                                  1 A000045
{1,-1,0,2,-3,-1,11,-15}                          -1 A007440

### Triangular numbers

{1,3,6,10,15,21,28,36}                                    1 A000217
{1,6,30,137,588,2415,9600,37209}                2 A030280
{1,-3,12,-55,273,-1428,7752,-43263}           -1 A001764

Converted by Mathematica      January 5, 2004