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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) ).

%I A357681 #26 Feb 16 2025 08:34:04
%S A357681 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,
%T A357681 30,70,97,0,1,0,6,15,44,135,330,434,0,1,0,7,18,60,220,705,1694,2095,0,
%U A357681 1,0,8,21,78,325,1228,3906,9202,10707,0,1,0,9,24,98,450,1905,7196,22953,53334,58194,0
%N 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) ).
%H A357681 Andrew Howroyd, <a href="/A357681/b357681.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A357681 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357681 T(n,k) = Sum_{j=0..floor(n/2)} k^j * Stirling2(n,2*j).
%F A357681 T(n,k) = ( Bell_n(sqrt(k)) + Bell_n(-sqrt(k)) )/2, where Bell_n(x) is n-th Bell polynomial.
%e A357681 Square array begins:
%e A357681   1,  1,  1,   1,   1,   1, ...
%e A357681   0,  0,  0,   0,   0,   0, ...
%e A357681   0,  1,  2,   3,   4,   5, ...
%e A357681   0,  3,  6,   9,  12,  15, ...
%e A357681   0,  8, 18,  30,  44,  60, ...
%e A357681   0, 25, 70, 135, 220, 325, ...
%o A357681 (PARI) T(n, k) = sum(j=0, n\2, k^j*stirling(n, 2*j, 2));
%o A357681 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357681 T(n, k) = round((Bell_poly(n, sqrt(k))+Bell_poly(n, -sqrt(k))))/2;
%Y A357681 Columns k=0-4 give: A000007, A024430, A264036, A357615, A065143.
%Y A357681 Column k=9 gives A357667.
%Y A357681 Main diagonal gives A357682.
%Y A357681 Cf. A292860.
%K A357681 nonn,tabl
%O A357681 0,13
%A A357681 _Seiichi Manyama_, Oct 09 2022