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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, 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 (list; table; graph; refs; listen; history; text; internal format)
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: A264704 A264659 A264560 * A264638 A201737 A080063

Adjacent sequences:  A300622 A300623 A300624 * A300626 A300627 A300628

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 12 2018

STATUS

approved

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Last modified May 28 15:04 EDT 2022. Contains 354115 sequences. (Running on oeis4.)