login
A357681
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) * (exp(x) - 1) ).
8
1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 8, 0, 1, 0, 4, 9, 18, 25, 0, 1, 0, 5, 12, 30, 70, 97, 0, 1, 0, 6, 15, 44, 135, 330, 434, 0, 1, 0, 7, 18, 60, 220, 705, 1694, 2095, 0, 1, 0, 8, 21, 78, 325, 1228, 3906, 9202, 10707, 0, 1, 0, 9, 24, 98, 450, 1905, 7196, 22953, 53334, 58194, 0
OFFSET
0,13
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Bell Polynomial.
FORMULA
T(n,k) = Sum_{j=0..floor(n/2)} k^j * Stirling2(n,2*j).
T(n,k) = ( Bell_n(sqrt(k)) + Bell_n(-sqrt(k)) )/2, where Bell_n(x) is n-th Bell polynomial.
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, 8, 18, 30, 44, 60, ...
0, 25, 70, 135, 220, 325, ...
PROG
(PARI) T(n, k) = sum(j=0, n\2, k^j*stirling(n, 2*j, 2));
(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = round((Bell_poly(n, sqrt(k))+Bell_poly(n, -sqrt(k))))/2;
CROSSREFS
Columns k=0-4 give: A000007, A024430, A264036, A357615, A065143.
Column k=9 gives A357667.
Main diagonal gives A357682.
Cf. A292860.
Sequence in context: A064722 A259748 A357728 * A357720 A357712 A298159
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 09 2022
STATUS
approved