A382101 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(B_k(x) - 1), where B_k(x) = 1 + x*B_k(x)^k.
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 43, 73, 1, 1, 1, 9, 91, 529, 501, 1, 1, 1, 11, 157, 1753, 8501, 4051, 1, 1, 1, 13, 241, 4129, 45001, 169021, 37633, 1, 1, 1, 15, 343, 8041, 146001, 1447471, 4010455, 394353, 1, 1, 1, 17, 463, 13873, 362501, 6502681, 56041987, 110676833, 4596553, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, ... 1, 3, 5, 7, 9, 11, ... 1, 13, 43, 91, 157, 241, ... 1, 73, 529, 1753, 4129, 8041, ... 1, 501, 8501, 45001, 146001, 362501, ...
Programs
-
PARI
a(n, k) = if(n==0, 1, (n-1)!*pollaguerre(n-1, (k-1)*n+1, -1));
Formula
A(n,k) = (n-1)! * Sum_{j=0..n-1} binomial(k*n,j)/(n-j-1)! for n > 0.
A(n,k) = (n-1)! * LaguerreL(n-1, (k-1)*n+1, -1) for n > 0.
E.g.f. of column k: exp( Series_Reversion( x/(1+x)^k ) ).