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A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).

%I A357293 #26 Feb 16 2025 08:34:04
%S A357293 1,1,0,1,1,0,1,0,2,0,1,0,1,5,0,1,0,0,3,15,0,1,0,0,1,8,52,0,1,0,0,0,6,
%T A357293 25,203,0,1,0,0,0,1,25,97,877,0,1,0,0,0,0,10,91,434,4140,0,1,0,0,0,0,
%U A357293 1,65,322,2095,21147,0,1,0,0,0,0,0,15,350,1232,10707,115975,0,1,0,0,0,0,0,1,140,1702,5672,58194,678570,0
%N A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).
%H A357293 Andrew Howroyd, <a href="/A357293/b357293.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A357293 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357293 For k > 0, e.g.f. of column k: Sum_{j>=0} (exp(x)-1)^(k*j)/(k*j)!.
%F A357293 For k > 0, T(n,k) = ( Sum_{j=0..k-1} Bell_n(w^j) )/k, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/k).
%e A357293 Square array begins:
%e A357293   1,   1,  1,  1,  1,  1, 1, ...
%e A357293   0,   1,  0,  0,  0,  0, 0, ...
%e A357293   0,   2,  1,  0,  0,  0, 0, ...
%e A357293   0,   5,  3,  1,  0,  0, 0, ...
%e A357293   0,  15,  8,  6,  1,  0, 0, ...
%e A357293   0,  52, 25, 25, 10,  1, 0, ...
%e A357293   0, 203, 97, 91, 65, 15, 1, ...
%o A357293 (PARI) T(n, k) = sum(j=0, n, stirling(n, k*j, 2));
%o A357293 (PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (exp(x+x*O(x^n))-1)^(k*j)/(k*j)!), n));
%o A357293 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357293 T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Bell_poly(n, w^j)))/k);
%Y A357293 Columns k=0-3 give: A000007, A000110, A024430, A143815.
%Y A357293 Cf. A357119.
%K A357293 nonn,tabl
%O A357293 0,9
%A A357293 _Seiichi Manyama_, Oct 17 2022