A300625 Table of row functions R(n,x) that satisfy: [x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1, n>=1, read by antidiagonals.
1, 1, 1, 1, 2, 3, 1, 4, 27, 14, 1, 8, 243, 736, 85, 1, 16, 2187, 40448, 30525, 621, 1, 32, 19683, 2351104, 12519125, 1715454, 5236, 1, 64, 177147, 142475264, 6153518125, 6111917748, 123198985, 49680, 1, 128, 1594323, 8856272896, 3436799053125, 31779658925496, 4308276119854, 10931897664, 521721, 1, 256, 14348907, 558312194048, 2049047412828125, 212148041589128016, 287364845865893467, 4151360558858752, 1172808994833, 5994155
Offset: 1
Examples
This table of coefficients T(n,k) begins:
n=1: [1, 1, 3, 14, 85, 621, 5236, 49680, ...];
n=2: [1, 2, 27, 736, 30525, 1715454, 123198985, 10931897664, ...];
n=3: [1, 4, 243, 40448, 12519125, 6111917748, 4308276119854, ..];
n=4: [1, 8, 2187, 2351104, 6153518125, 31779658925496, ...];
n=5: [1, 16, 19683, 142475264, 3436799053125, 212148041589128016, ...];
n=6: [1, 32, 177147, 8856272896, 2049047412828125, 1569837215111038900704, ...];
n=7: [1, 64, 1594323, 558312194048, 1256793474918203125, 12020665333382306853887808, ...]; ...
such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
[x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1.
Row functions R(n,x) begin:
R(1,x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 + ...
R(2,x) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + ...
R(3,x) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + ...
R(4,x) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + ...
etc.
Links
Programs
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PARI
{T(n, k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^n)); A[#A] = ((#A-1)^n * V[#A-1] - V[#A])/(#A-1)^n ); polcoeff( log(Ser(A)), k)} /* Print as a table of row functions: */ for(n=1, 8, for(k=1, 8, print1(T(n, k), ", ")); print("")) /* Print as a flattened triangle: */ for(n=1, 12, for(k=1, n-1, print1(T(n-k, k), ", ")); )