A156305 G.f.: A(x) = exp( Sum_{n>=1} sigma(n) * C(2*n-1,n) * x^n/n ), a power series in x with integer coefficients.

1, 1, 5, 18, 87, 290, 1553, 5015, 25436, 94500, 431464, 1519749, 8024004, 26746757, 125190249, 498138920, 2221127601, 8020960187, 38836436844, 138444409552, 655009491676, 2512996318026, 10775473291178, 40824090856703

Offset

0,3

Comments

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ),

and to the g.f. of Catalan numbers: exp( Sum_{n>=1} C(2*n-1,n)*x^n/n ),

where sigma(n) = A000203(n) is the sum of the divisors of n.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k)*C(2*k-1,k)*a(n-k) for n>0, with a(0) = 1.

Example

G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 87*x^4 + 290*x^5 + 1553*x^6 + 5015*x^7 + ...
log(A(x)) = x + 3*3*x^2/2 + 4*10*x^3/3 + 7*35*x^4/4 + 6*126*x^5/5 + 12*462*x^6/6 + ... + A000203(n)*A001700(n)*x^n/n + ...

Mathematica

a[n_] := If[n==0, 1, (1/n) * Sum[DivisorSigma[1, k] * Binomial[2k - 1, k] a[n - k], {k, n}] ]; Table[a[n], {n, 0, 23}] (* Indranil Ghosh, Mar 12 2017 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)*binomial(2*k-1, k)*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)*binomial(2*k-1, k)*a(n-k)))}

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A001700 (C(2*n-1, n)), A000108 (Catalan).

Sequence in context: A357797 A199257 A188206 * A369362 A213190 A151490

Adjacent sequences: A156302 A156303 A156304 * A156306 A156307 A156308

Keyword

nonn

Author

Paul D. Hanna, Feb 08 2009

Status

approved