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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.

%I A382101 #19 Mar 16 2025 12:42:09
%S A382101 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,
%T A382101 11,157,1753,8501,4051,1,1,1,13,241,4129,45001,169021,37633,1,1,1,15,
%U A382101 343,8041,146001,1447471,4010455,394353,1,1,1,17,463,13873,362501,6502681,56041987,110676833,4596553,1
%N 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.
%H A382101 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A382101 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A382101 A(n,k) = (n-1)! * Sum_{j=0..n-1} binomial(k*n,j)/(n-j-1)! for n > 0.
%F A382101 A(n,k) = (n-1)! * LaguerreL(n-1, (k-1)*n+1, -1) for n > 0.
%F A382101 E.g.f. of column k: exp( Series_Reversion( x/(1+x)^k ) ).
%e A382101 Square array begins:
%e A382101   1,   1,    1,     1,      1,      1, ...
%e A382101   1,   1,    1,     1,      1,      1, ...
%e A382101   1,   3,    5,     7,      9,     11, ...
%e A382101   1,  13,   43,    91,    157,    241, ...
%e A382101   1,  73,  529,  1753,   4129,   8041, ...
%e A382101   1, 501, 8501, 45001, 146001, 362501, ...
%o A382101 (PARI) a(n, k) = if(n==0, 1, (n-1)!*pollaguerre(n-1, (k-1)*n+1, -1));
%Y A382101 Columns k=0..4 give A000012, A000262, A251568, A380512, A380516.
%Y A382101 Cf. A355262, A382100.
%K A382101 nonn,tabl
%O A382101 0,9
%A A382101 _Seiichi Manyama_, Mar 15 2025