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A357712 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * log(1-x) ).

%I A357712 #18 Feb 16 2025 08:34:04
%S A357712 1,1,0,1,0,0,1,0,1,0,1,0,2,3,0,1,0,3,6,12,0,1,0,4,9,26,60,0,1,0,5,12,
%T A357712 42,140,360,0,1,0,6,15,60,240,896,2520,0,1,0,7,18,80,360,1614,6636,
%U A357712 20160,0,1,0,8,21,102,500,2520,12474,55804,181440,0,1,0,9,24,126,660,3620,20160,108900,525168,1814400,0
%N A357712 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * log(1-x) ).
%H A357712 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357712 T(n,k) = Sum_{j=0..floor(n/2)} k^j * |Stirling1(n,2*j)|.
%F A357712 T(n,k) = ( (sqrt(k))_n + (-sqrt(k))_n )/2, where (x)_n is the Pochhammer symbol.
%F A357712 T(0,k) = 1, T(1,k) = 0; T(n,k) = (2*n-3) * T(n-1,k) - (n^2-4*n+4-k) * T(n-2,k).
%e A357712 Square array begins:
%e A357712   1,  1,   1,   1,   1,   1, ...
%e A357712   0,  0,   0,   0,   0,   0, ...
%e A357712   0,  1,   2,   3,   4,   5, ...
%e A357712   0,  3,   6,   9,  12,  15, ...
%e A357712   0, 12,  26,  42,  60,  80, ...
%e A357712   0, 60, 140, 240, 360, 500, ...
%o A357712 (PARI) T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));
%o A357712 (PARI) T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;
%Y A357712 Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.
%Y A357712 Main diagonal gives A357683.
%Y A357712 Cf. A357681.
%K A357712 nonn,tabl
%O A357712 0,13
%A A357712 _Seiichi Manyama_, Oct 10 2022