A357552 a(n) = sigma(n) * binomial(2*n-1,n), for n >= 1.

1, 9, 40, 245, 756, 5544, 13728, 96525, 316030, 1662804, 4232592, 37858184, 72804200, 481399200, 1861410240, 9316746045, 21002455980, 176965138350, 353452638000, 2894777105220, 8612125991040, 37873781346960, 98801168731200, 967428110493000, 1959364399785156

Offset: 1

Author

Paul D. Hanna, Nov 14 2022

Keywords

• nonn

Comments

Equals the coefficients in the logarithmic derivative of the g.f. of A156305.

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.

Crossrefs

Cf. A000203 (sigma(n)), A001700 (C(2*n-1, n)), A156305, A158267, A225528.

Programs

Mathematica

Table[DivisorSigma[1,n]Binomial[2n-1,n],{n,30}] (* Harvey P. Dale, Aug 19 2025 *)

PARI

{a(n) = sigma(n) * binomial(2*n-1,n)}
for(n=1,30,print1(a(n),", "))

Formula

L.g.f.: L(x) = x + 9*x^2/2 + 40*x^3/3 + 245*x^4/4 + 756*x^5/5 + 5544*x^6/6 + 13728*x^7/7 + 96525*x^8/8 + 316030*x^9/9 + 1662804*x^10/10 + 4232592*x^11/11 + 37858184*x^12/12 + ... + a(n)*x^n/n + ...

equivalently,

L(x) = 1*1*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 + 8*1716*x^7/7 + 15*6435*x^8/8 + ... + sigma(n)*binomial(2*n-1,n)*x^n/n + ...

where exponentiation yields the integer series given by A156305:

exp(L(x)) = 1 + x + 5*x^2 + 18*x^3 + 87*x^4 + 290*x^5 + 1553*x^6 + 5015*x^7 + 25436*x^8 + 94500*x^9 + 431464*x^10 + ... + A156305(n)*x^n + ...