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A264036 Stirling transform of A077957 (aerated powers of 2).

%I A264036 #22 Feb 16 2025 08:33:27
%S A264036 1,0,2,6,18,70,330,1694,9202,53334,332090,2212782,15638370,116365990,
%T A264036 907975146,7413080510,63212284498,561747543414,5190343710746,
%U A264036 49752410984526,493844719701186,5068209425457862,53705511911500746,586862875255860062,6605213319604075186
%N A264036 Stirling transform of A077957 (aerated powers of 2).
%C A264036 a(n) is the inverse binomial transform of A264037 without the leading zero [1, 1, 3, 13, 55, ...].
%H A264036 Seiichi Manyama, <a href="/A264036/b264036.txt">Table of n, a(n) for n = 0..567</a>
%H A264036 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A264036 a(n) = Sum_{k=0..n} A077957(k)*Stirling2(n,k).
%F A264036 a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k).
%F A264036 a(n) = (Bell_n(sqrt(2)) + Bell_n(-sqrt(2)))/2, where Bell_n(x) is n-th Bell polynomial.
%F A264036 Bell_n(sqrt(2)) = a(n) + A264037(n)*sqrt(2).
%F A264036 E.g.f.: cosh(sqrt(2)*(exp(x) - 1)).
%F A264036 a(n) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k) * A264037(k). - _Seiichi Manyama_, Oct 12 2022
%e A264036 G.f. = 1 + 2*x^2 + 6*x^3 + 18*x^4 + 70*x^5 + 330*x^6 + 1694*x^7 + 9202*x^8 + ...
%t A264036 Table[(BellB[n, Sqrt[2]] + BellB[n, -Sqrt[2]])/2, {n, 0, 24}]
%o A264036 (PARI) vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k, 2))) \\ _Altug Alkan_, Nov 01 2015
%Y A264036 Column k=2 of A357681.
%Y A264036 Cf. A065143, A077957, A264037.
%K A264036 nonn
%O A264036 0,3
%A A264036 _Vladimir Reshetnikov_, Nov 01 2015