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

%I A357552 #15 Aug 19 2025 17:29:41
%S A357552 1,9,40,245,756,5544,13728,96525,316030,1662804,4232592,37858184,
%T A357552 72804200,481399200,1861410240,9316746045,21002455980,176965138350,
%U A357552 353452638000,2894777105220,8612125991040,37873781346960,98801168731200,967428110493000,1959364399785156
%N A357552 a(n) = sigma(n) * binomial(2*n-1,n), for n >= 1.
%C A357552 Equals the coefficients in the logarithmic derivative of the g.f. of A156305.
%C A357552 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.
%F A357552 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 + ...
%F A357552 equivalently,
%F A357552 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 + ...
%F A357552 where exponentiation yields the integer series given by A156305:
%F A357552 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 + ...
%t A357552 Table[DivisorSigma[1,n]Binomial[2n-1,n],{n,30}] (* _Harvey P. Dale_, Aug 19 2025 *)
%o A357552 (PARI) {a(n) = sigma(n) * binomial(2*n-1,n)}
%o A357552 for(n=1,30,print1(a(n),", "))
%Y A357552 Cf. A000203 (sigma(n)), A001700 (C(2*n-1, n)), A156305, A158267, A225528.
%K A357552 nonn
%O A357552 1,2
%A A357552 _Paul D. Hanna_, Nov 14 2022