A156733 - OEIS

%I #18 Nov 03 2020 04:56:56

%S 1,1,11,41,176,606,2391,8091,28636,95056,316048,1014240,3237325,

%T 10082015,31109500,94352346,283209381,838650191,2458835711,7127912979,

%U 20471486368,58224189612,164181018330,458982667630,1273039111210,3503609456548,9572771822745,25971150308985

%N Euler transform of n*A065958(n).

%C Compare to the g.f. of planar partitions (A000219): exp( Sum_{n>=1} sigma(n,2)*x^n/n ) = Product_{n>=1} 1/(1-x^n)^n.

%H Alois P. Heinz, <a href="/A156733/b156733.txt">Table of n, a(n) for n = 0..2000</a>

%F a(n) = (1/n)*Sum_{k=1..n} sigma_2(k^2)*a(n-k) for n>0, with a(0) = 1.

%F G.f.: exp( Sum_{n>=1} A065827(n)*x^n/n ), where A065827(n) = sigma_2(n^2) is the sum of squares of the divisors of n^2. - _Paul D. Hanna_, Aug 09 2012

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       a(n-j)*numtheory[sigma][2](j^2), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 24 2016

%t a[0] = 1;

%t a[n_] := a[n] = (1/n) Sum[DivisorSigma[2, k^2] a[n-k], {k, 1, n}];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 03 2020 *)

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(m^2,2)*x^m/m)+x*O(x^n)),n)}

%o for(n=0,21,print1(a(n),", "))

%Y Cf. A001157, A156303, A065827, A301978, A301980.

%K nonn

%O 0,3

%A _Paul D. Hanna_ and _Vladeta Jovovic_, Feb 14 2009