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

1,5

Links

Paul D. Hanna, Table of n, a(n) for n = 1..435 of rows 1..30 as a flattened table read by antidiagonals.

Example

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.

Prog

(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), ", ")); )

Crossrefs

Cf. A088716 (row 1), A300591 (row 2), A300595 (row 3), A300597 (row 4).

Sequence in context: A264659 A371295 A264560 * A264638 A201737 A080063

Adjacent sequences: A300622 A300623 A300624 * A300626 A300627 A300628

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 12 2018

Status

approved