A156305 -id:A156305 - cp's OEIS frontend

A158267 Inverse Euler transform of A156305.

1, 4, 13, 59, 151, 916, 1961, 12035, 35110, 166204, 384781, 3154367, 5600323, 34384676, 124093963, 582290595, 1235438587, 9831378712, 18602770421, 144738772109, 410101237013, 1721535323380, 4295702988313, 40309503022439

Offset: 1

Paul D. Hanna, Apr 09 2009

nonn

G.f. of A156305: exp( Sum_{n>=1} sigma(n)*C(2*n-1,n)*x^n/n ), where C(2n-1,n) = A001700(n-1).

			Let G(x) = g.f. of A156305:
G(x) = 1 + x + 5*x^2 + 18*x^3 + 87*x^4 + 290*x^5 + 1553*x^6 +...
G(x) = 1/[(1-x)*(1-x^2)^4*(1-x^3)^13*(1-x^4)^59*(1-x^5)^151*...].

Cf. A156305, A001700.

Table[Sum[DivisorSigma[1, d]*Binomial[2*d - 1, d]*MoebiusMu[n/d], {d, Divisors[n]}] / n, {n, 1, 30}] (* Vaclav Kotesovec, Oct 09 2019 *)

{a(n)=(1/n)*sumdiv(n,d, sigma(d)*binomial(2*d-1, d)*moebius(n/d))}

a(n) = (1/n)*Sum_{d|n} sigma(d)*C(2d-1,d)*moebius(n/d).

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

Original entry on oeis.org

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

Paul D. Hanna, Nov 14 2022

nonn

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.

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

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

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

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 + ...

cp's OEIS Frontend

A158267 Inverse Euler transform of A156305.

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