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A357720 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * log(1+x) ).

%I A357720 #12 Feb 16 2025 08:34:04
%S A357720 1,1,0,1,0,0,1,0,-1,0,1,0,-2,3,0,1,0,-3,6,-10,0,1,0,-4,9,-18,40,0,1,0,
%T A357720 -5,12,-24,60,-190,0,1,0,-6,15,-28,60,-216,1050,0,1,0,-7,18,-30,40,
%U A357720 -84,756,-6620,0,1,0,-8,21,-30,0,200,-756,-1620,46800,0,1,0,-9,24,-28,-60,630,-3360,13104,-14256,-365300,0
%N A357720 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * log(1+x) ).
%H A357720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357720 T(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * Stirling1(n,2*j).
%F A357720 T(n,k) = (-1)^n * ( (sqrt(k) * i)_n + (-sqrt(k) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
%F A357720 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 A357720 Square array begins:
%e A357720   1,   1,   1,   1,   1,   1, ...
%e A357720   0,   0,   0,   0,   0,   0, ...
%e A357720   0,  -1,  -2,  -3,  -4,  -5, ...
%e A357720   0,   3,   6,   9,  12,  15, ...
%e A357720   0, -10, -18, -24, -28, -30, ...
%e A357720   0,  40,  60,  60,  40,   0, ...
%o A357720 (PARI) T(n, k) = sum(j=0, n\2, (-k)^j*stirling(n, 2*j, 1));
%o A357720 (PARI) T(n, k) = (-1)^n*round((prod(j=0, n-1, sqrt(k)*I+j)+prod(j=0, n-1, -sqrt(k)*I+j)))/2;
%Y A357720 Columns k=0-4 give: A000007, (-1)^n * A003703, A357693, A357718, A357719.
%Y A357720 Main diagonal gives A357721.
%Y A357720 Cf. A357712, A357728.
%K A357720 sign,tabl
%O A357720 0,13
%A A357720 _Seiichi Manyama_, Oct 10 2022