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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. 9

%I

%S 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,

%T 466241,22417,1,1,1,1985,2246281,153143499,162016980,685592,1,1,1,

%U 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

%N 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.

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

%H Paul D. Hanna, <a href="/A304320/b304320.txt">Table of n, a(n) for n = 1..1326 as a flattened table read by antidiagonals 1..51.</a>

%F 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

%e This table begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;

%e 1, 1, 5, 54, 935, 22417, 685592, 25431764, 1106630687, 55174867339, ...;

%e 1, 1, 25, 2317, 466241, 162016980, 85975473871, 64545532370208, ...;

%e 1, 1, 113, 76446, 153143499, 673638499100, 5510097691767062, ...;

%e 1, 1, 481, 2246281, 43087884081, 2331601789103231, 287133439746933073357, ...;

%e 1, 1, 1985, 62861994, 11442690973075, 7570836550478960487, ...;

%e 1, 1, 8065, 1723380877, 2972352315820441, 24013530904194819396970, ...;

%e 1, 1, 32513, 46836819846, 765428206086770699, 75487364859452767380638650, ...;

%e 1, 1, 130561, 1268169652561, 196425341268811084961, 236460748444613412476233431261, ...; ...

%e 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.

%o (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]}

%o /* Print table: */

%o for(n=1,8, for(k=0,8, print1( T(n,k),", "));print(""))

%o /* Print as a flattened table: */

%o for(n=0,10, for(k=0,n, print1( T(n-k+1,k),", "));)

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

%K nonn

%O 1,9

%A _Paul D. Hanna_, May 11 2018

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Last modified March 8 11:48 EST 2021. Contains 341948 sequences. (Running on oeis4.)