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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) ).
3
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, 42, 140, 360, 0, 1, 0, 6, 15, 60, 240, 896, 2520, 0, 1, 0, 7, 18, 80, 360, 1614, 6636, 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
OFFSET
0,13
LINKS
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
FORMULA
T(n,k) = Sum_{j=0..floor(n/2)} k^j * |Stirling1(n,2*j)|.
T(n,k) = ( (sqrt(k))_n + (-sqrt(k))_n )/2, where (x)_n is the Pochhammer symbol.
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).
EXAMPLE
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, 12, 26, 42, 60, 80, ...
0, 60, 140, 240, 360, 500, ...
PROG
(PARI) T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));
(PARI) T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;
CROSSREFS
Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.
Main diagonal gives A357683.
Cf. A357681.
Sequence in context: A357728 A357681 A357720 * A298159 A123735 A155839
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 10 2022
STATUS
approved