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A305562 Coefficients associated with power series solution of g'(x) = g(x/2) e^x at x=0.

%I A305562 #22 Jul 21 2025 04:59:37
%S A305562 1,1,3,19,251,6843,381851,43357211,9976746651,4639483488923,
%T A305562 4351708606681243,8221479626141796507,31252321079882850259099,
%U A305562 238835886863534101328335003,3667031594654877566958673359003,113055325655546855868908521812586651
%N A305562 Coefficients associated with power series solution of g'(x) = g(x/2) e^x at x=0.
%C A305562 The series g(x) := Sum_{n>=0} a(n) * 2^((-n*n+n)/2) * x^n / n! satisfies g'(x) = g(x/2) e^x.
%C A305562 The denominators of the power series can be found at A006125. - _Lewis Chen_, Apr 28 2019
%H A305562 FiveThirtyEight, <a href="https://fivethirtyeight.com/features/how-many-earthlings-would-survive-63-thanos-snaps/">The Riddler (Apr. 26, 2019)</a> (Riddler Express), with <a href="https://fivethirtyeight.com/features/are-you-the-best-warlord/">solution</a>
%F A305562 a(n+1) = Sum_{n=0..k} a(k)*binomial(n, k)*2^( (n*n + n - k*k-k)/2 ).
%F A305562 Conjecture: a(n) = A386393(n,n) for n > 0. - _Mikhail Kurkov_, Jul 19 2025
%o A305562 (PARI) {a(n) = if( n<1, n==0, n--; sum(k=0, n, a(k) * binomial(n, k) * 2^( (n*n+n - k*k-k)/2 )))};
%Y A305562 Cf. A006125, A386393.
%K A305562 nonn,frac
%O A305562 0,3
%A A305562 _Michael Somos_, Jun 04 2018