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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) ).
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%I #17 Oct 10 2022 10:15:29

%S 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 42,140,360,0,1,0,6,15,60,240,896,2520,0,1,0,7,18,80,360,1614,6636,

%U 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.

%F T(n,k) = Sum_{j=0..floor(n/2)} k^j * |Stirling1(n,2*j)|.

%F T(n,k) = ( (sqrt(k))_n + (-sqrt(k))_n )/2, where (x)_n is the Pochhammer symbol.

%F 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 Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 0, 0, 0, 0, 0, ...

%e 0, 1, 2, 3, 4, 5, ...

%e 0, 3, 6, 9, 12, 15, ...

%e 0, 12, 26, 42, 60, 80, ...

%e 0, 60, 140, 240, 360, 500, ...

%o (PARI) T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));

%o (PARI) T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;

%Y Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.

%Y Main diagonal gives A357683.

%Y Cf. A357681.

%K nonn,tabl

%O 0,13

%A _Seiichi Manyama_, Oct 10 2022