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) ).
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, -5, 12, -24, 60, -190, 0, 1, 0, -6, 15, -28, 60, -216, 1050, 0, 1, 0, -7, 18, -30, 40, -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
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 0, 0, 0, 0, 0, 0, ... 0, -1, -2, -3, -4, -5, ... 0, 3, 6, 9, 12, 15, ... 0, -10, -18, -24, -28, -30, ... 0, 40, 60, 60, 40, 0, ...
Links
- Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Crossrefs
Programs
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PARI
T(n, k) = sum(j=0, n\2, (-k)^j*stirling(n, 2*j, 1));
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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;
Formula
T(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * Stirling1(n,2*j).
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.
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).