A304320 Table of coefficients in row functions R(n,x) such that [x^k] exp( k^n * x ) / R(n,x) = 0 for k>=1 and n>=1.

1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 25, 54, 1, 1, 1, 113, 2317, 935, 1, 1, 1, 481, 76446, 466241, 22417, 1, 1, 1, 1985, 2246281, 153143499, 162016980, 685592, 1, 1, 1, 8065, 62861994, 43087884081, 673638499100, 85975473871, 25431764, 1, 1, 1, 32513, 1723380877, 11442690973075, 2331601789103231, 5510097691767062, 64545532370208, 1106630687, 1, 1, 1, 130561, 46836819846, 2972352315820441, 7570836550478960487, 287133439746933073357, 75312181798660695788, 65062315637060121, 55174867339, 1

Offset

1,9

Comments

It is striking that the coefficients in this table consist entirely of integers.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1326 as a flattened table read by antidiagonals 1..51.

Formula

For fixed row r > 1 is a(n) ~ sqrt(1-c) * r^(r*n) * n^((r-1)*n - 1/2) / (sqrt(2*Pi) * c^n * (r-c)^((r-1)*n) * exp((r-1)*n)), where c = -LambertW(-r*exp(-r)). - Vaclav Kotesovec, Aug 31 2020

Example

This table begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 1, 5, 54, 935, 22417, 685592, 25431764, 1106630687, 55174867339, ...;
1, 1, 25, 2317, 466241, 162016980, 85975473871, 64545532370208, ...;
1, 1, 113, 76446, 153143499, 673638499100, 5510097691767062, ...;
1, 1, 481, 2246281, 43087884081, 2331601789103231, 287133439746933073357, ...;
1, 1, 1985, 62861994, 11442690973075, 7570836550478960487, ...;
1, 1, 8065, 1723380877, 2972352315820441, 24013530904194819396970, ...;
1, 1, 32513, 46836819846, 765428206086770699, 75487364859452767380638650, ...;
1, 1, 130561, 1268169652561, 196425341268811084961, 236460748444613412476233431261, ...; ...
Let R(n,x) denote the o.g.f. of row n of this table, then the coefficient of x^k in exp(k^n*x)/R(n,x) = 0 for k>=1 and n>=1.

Prog

(PARI) {T(n, k) = my(A=[1], m); for(i=1, k, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^n +x*O(x^m)) / Ser(A) )[m] ); A[k+1]}
/* Print table: */
for(n=1, 8, for(k=0, 8, print1( T(n, k), ", ")); print(""))
/* Print as a flattened table: */
for(n=0, 10, for(k=0, n, print1( T(n-k+1, k), ", ")); )

Crossrefs

Cf. A304321, A304322 (row 2), A304323 (row 3), A304324 (row 4), A304325 (row 5), A337551 (diagonal).

Sequence in context: A230368 A256690 A181985 * A130511 A320410 A011396

Adjacent sequences: A304317 A304318 A304319 * A304321 A304322 A304323

Keyword

nonn,tabl

Author

Paul D. Hanna, May 11 2018

Status

approved