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

1, 1, 3, 19, 251, 6843, 381851, 43357211, 9976746651, 4639483488923, 4351708606681243, 8221479626141796507, 31252321079882850259099, 238835886863534101328335003, 3667031594654877566958673359003, 113055325655546855868908521812586651

Offset: 0

Author

Michael Somos, Jun 04 2018

Keywords

• nonn
• frac

Comments

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.

The denominators of the power series can be found at A006125. - Lewis Chen, Apr 28 2019

Links

• FiveThirtyEight, The Riddler (Apr. 26, 2019) (Riddler Express), with solution

Crossrefs

Cf. A006125, A386393.

Programs

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 )))};

Formula

a(n+1) = Sum_{n=0..k} a(k)*binomial(n, k)*2^( (n*n + n - k*k-k)/2 ).

Conjecture: a(n) = A386393(n,n) for n > 0. - Mikhail Kurkov, Jul 19 2025