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A357119 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} |Stirling1(n,k*j)|.

%I A357119 #19 Feb 16 2025 08:34:04
%S A357119 1,1,0,1,1,0,1,0,2,0,1,0,1,6,0,1,0,0,3,24,0,1,0,0,1,12,120,0,1,0,0,0,
%T A357119 6,60,720,0,1,0,0,0,1,35,360,5040,0,1,0,0,0,0,10,226,2520,40320,0,1,0,
%U A357119 0,0,0,1,85,1645,20160,362880,0,1,0,0,0,0,0,15,735,13454,181440,3628800,0
%N A357119 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} |Stirling1(n,k*j)|.
%H A357119 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357119 For k > 0, e.g.f. of column k: Sum_{j>=0} (-log(1-x))^(k*j)/(k*j)!.
%F A357119 For k > 0, T(n,k) = ( Sum_{j=0..k-1} (w^j)_n )/k, where (x)_n is the Pochhammer symbol and w = exp(2*Pi*i/k).
%e A357119 Square array begins:
%e A357119   1,   1,   1,   1,  1,  1, 1, ...
%e A357119   0,   1,   0,   0,  0,  0, 0, ...
%e A357119   0,   2,   1,   0,  0,  0, 0, ...
%e A357119   0,   6,   3,   1,  0,  0, 0, ...
%e A357119   0,  24,  12,   6,  1,  0, 0, ...
%e A357119   0, 120,  60,  35, 10,  1, 0, ...
%e A357119   0, 720, 360, 226, 85, 15, 1, ...
%o A357119 (PARI) T(n, k) = sum(j=0, n, abs(stirling(n, k*j, 1)));
%o A357119 (PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (-log(1-x+x*O(x^n)))^(k*j)/(k*j)!), n));
%o A357119 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o A357119 T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Pochhammer(w^j, n)))/k);
%Y A357119 Columns k=0-3 give: A000007, A000142, (-1)^n * A105752(n), A357828.
%Y A357119 Cf. A357293.
%K A357119 nonn,tabl
%O A357119 0,9
%A A357119 _Seiichi Manyama_, Oct 17 2022