A396159 Numerators in the asymptotic expansion of Gamma(1 + 2*x)/(Gamma(1 + x)*Gamma(2 + x)) = (4^x/sqrt(Pi))*Sum_{k>=0} (1/x)^(3/2+k)*a(k)/A061549(k).

1, -9, 145, -1155, 36939, -295911, 4735445, -37844235, 2421696563, -19402289907, 310496335695, -2475570313125, 79183820728095, -640669821436755, 10265348737564125, -77871582933450075, 9898775194885242435, -105666176941167049515, 1744050969382356538475, 12322967114104483154175

Offset

0,2

Comments

Gamma(1 + 2*n)/(Gamma(1 + n)*Gamma(2 + n)) interpolates the Catalan numbers A000108.

Links

Table of n, a(n) for n=0..19.

Neven Elezović, Asymptotic expansions of gamma and related functions, binomial coefficients, inequalities and means, Journal of Mathematical Inequalities, Volume 9, Number 4 (2015), 1001-1054.

Formula

a(n) = numerator( (1/n)*Sum_{k=1..n} ( ((2^(-k)-2)*B_{k+1}/(k+1)) + (-1)^k )*a(n-k)/A061549(n-k) ), for n > 0 with a(0) = 1 where B_{k} is the Bernoulli number A027641(k)/A027642(k).

abs(A143503(n+2)) = abs(abs(a(n+1)) - abs(a(n))*A061549(n+1)/A061549(n)).

Prog

(PARI)
p(m) = if(m==0, 1, (1/m)*sum(k=1, m, (((2^(-k)-2)*bernfrac(k+1)/(k+1))+(-1)^k)*p(m-k)))
a(n) = numerator(p(n))

Crossrefs

Cf. A000108, A027641, A027642, A143503.

Cf. A061549 (denominators).

Sequence in context: A094594 A173213 A223371 * A046529 A331329 A388729

Adjacent sequences: A396156 A396157 A396158 * A396160 A396161 A396162

Keyword

sign,frac

Author

Thomas Scheuerle, May 18 2026

Status

approved