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

0,3

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

Table of n, a(n) for n=0..15.

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

Formula

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

Prog

(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 )))};

Crossrefs

Cf. A006125.

Sequence in context: A054590 A261495 A069344 * A356672 A316294 A233240

Adjacent sequences: A305559 A305560 A305561 * A305563 A305564 A305565

Keyword

nonn,frac

Author

Michael Somos, Jun 04 2018

Status

approved