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A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].

%I A264037 #21 Feb 16 2025 08:33:27
%S A264037 0,1,1,3,13,55,241,1171,6357,37567,236521,1574331,11068333,82110535,
%T A264037 640794337,5239439011,44723250501,397481121295,3671081354137,
%U A264037 35176098791115,349120380267421,3583273413146647,37975511840454673,415004245048757299,4670891190907818165
%N A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].
%C A264037 a(n) without the leading zero [1, 1, 3, 13, 55, ...] is the binomial transform of A264036.
%H A264037 Seiichi Manyama, <a href="/A264037/b264037.txt">Table of n, a(n) for n = 0..567</a>
%H A264037 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A264037 a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k+1).
%F A264037 a(n) = (Bell_n(sqrt(2)) - Bell_n(-sqrt(2)))/(2*sqrt(2)), where Bell_n(x) is n-th Bell polynomial.
%F A264037 Bell_n(sqrt(2)) = A264036(n) + a(n)*sqrt(2).
%F A264037 E.g.f.: sinh(sqrt(2)*(exp(x) - 1))/sqrt(2).
%F A264037 a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A264036(k). - _Seiichi Manyama_, Oct 12 2022
%e A264037 G.f. = x + x^2 + 3*x^3 + 13*x^4 + 55*x^5 + 241*x^7 + 1171*x^8 + 6357*x^9 + ...
%t A264037 Table[(BellB[n, Sqrt[2]] - BellB[n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 24}]
%o A264037 (PARI) vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k+1, 2))) \\ _Altug Alkan_, Nov 01 2015
%Y A264037 Cf. A077957, A264036.
%Y A264037 Cf. A024429, A357572, A357598.
%K A264037 nonn
%O A264037 0,4
%A A264037 _Vladimir Reshetnikov_, Nov 01 2015