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A277380 a(n) = Sum_{k>=1} H_n(k-1)/2^k, where H_n(x) is n-th Hermite polynomial.

%I A277380 #32 Feb 16 2025 08:33:36
%S A277380 1,2,10,92,1068,15352,265752,5368400,123919248,3217983008,92851377312,
%T A277380 2947037232064,102040223376576,3827536020146048,154615082607931776,
%U A277380 6691872388083371264,308938595472492867840,15153942107317778727424,787050616613300039649792
%N A277380 a(n) = Sum_{k>=1} H_n(k-1)/2^k, where H_n(x) is n-th Hermite polynomial.
%H A277380 G. C. Greubel, <a href="/A277380/b277380.txt">Table of n, a(n) for n = 0..380</a>
%H A277380 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>
%H A277380 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>
%F A277380 a(n) ~ c * 2^n * n! / (log(2))^n, where c = 1/(2 * log(2) * exp(log(2)^2/4)) = 0.639705404891769467944095575437073306645289161842121830191257596548619914238... - _Vaclav Kotesovec_, Jul 13 2018, updated Apr 21 2024
%t A277380 Table[Sum[HermiteH[n, k - 1]/2^k, {k, 1, Infinity}], {n, 0, 20}]
%o A277380 (PARI) for(n=0,40, print1(if(n==0,1, ceil(sum(k=1, 15*n, polhermite(n, k-1)/2^k))), ", ")) \\ _G. C. Greubel_, Jul 13 2018
%o A277380 (PARI) nmax = 40; p = floor(2*log(nmax!*(2/log(2))^nmax)/log(10)); default(realprecision, p); a(n) = round(suminf(k=1, polhermite(n, k-1)/2^k));
%o A277380 for(n=0, nmax, print1(a(n), ", ")); \\ _Michel Marcus_ and _Vaclav Kotesovec_, Jul 13 2018
%Y A277380 Cf. A277381, A316778.
%K A277380 nonn
%O A277380 0,2
%A A277380 _Vladimir Reshetnikov_, Oct 11 2016