See Schroeder Summations - Experimental Results for notes on limitations and implementation. Uses the Iterate.m package.
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}
{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
{1,1,2,3,5,8,13,21} 1 A000045
{1,-1,0,2,-3,-1,11,-15} -1 A007440
{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