A203321 - OEIS

%I #9 Dec 25 2023 18:20:04

%S 1,3,7,19,26,75,78,211,241,518,463,1447,1002,2558,2612,5715,3928,

%T 11901,7316,21574,17031,35159,23047,80575,40951,108488,86911,206638,

%U 107823,370220,173725,570803,372181,816496,451883,1723741,665150,2048982,1404150,3705366,1530859,5892479

%N L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n*k)*x^(n*k)/k ).

%H Paul D. Hanna, <a href="/A203321/b203321.txt">Table of n, a(n) for n = 1..120</a>

%F L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * P_n(x^n) where P_n(x^n) = Product_{k=0..n-1} P(u^k*x) where u is an n-th root of unity, and P(x) is the partition function (A000041); P(x) = exp(Sum_{n>=1} sigma(n)*x^n/n) where sigma(n) is the sum of divisors of n (A000203).

%e L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 75*x^6/6 +...

%e where

%e L(x) = x*exp(1*x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 +...) +

%e x^2/2*exp(3*x^2 + 7*x^4/2 + 12*x^6/3 + 15*x^8/4 + 18*x^10/5 +...) +

%e x^3/3*exp(4*x^3 + 12*x^6/2 + 13*x^9/3 + 28*x^12/4 + 24*x^15/5 +...) +

%e x^4/4*exp(7*x^4 + 15*x^8/2 + 28*x^12/3 + 31*x^16/4 + 42*x^20/5 +...) +

%e x^5/5*exp(6*x^5 + 18*x^10/2 + 24*x^15/3 + 42*x^20/4 + 31*x^25/5 +...) +

%e x^6/6*exp(12*x^6 + 28*x^12/2 + 39*x^18/3 + 60*x^24/4 + 72*x^30/5 +...) +

%e x^7/7*exp(8*x^7 + 24*x^14/2 + 32*x^21/3 + 56*x^28/4 + 48*x^35/5 +...) +

%e x^8/8*exp(15*x^8 + 31*x^16/2 + 60*x^24/3 + 63*x^32/4 + 90*x^40/5 +...) +...

%e ...

%e Equivalently, L(x) = Sum_{n>=1} P_n(x^n) * x^n/n where

%e P_n(x) = exp( Sum_{k>=1} sigma(n*k)*x^k/k ), which begin:

%e P_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...;

%e P_2(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...;

%e P_3(x) = 1 + 4*x + 14*x^2 + 39*x^3 + 101*x^4 + 238*x^5 + 533*x^6 +...;

%e P_4(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 + 3057*x^6 +...;

%e P_5(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + 2486*x^6 +...;

%e P_6(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 +...;

%e P_7(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 +...;

%e P_8(x) = 1 + 15*x + 128*x^2 + 815*x^3 + 4289*x^4 + 19663*x^5 +...;

%e ...

%o (PARI) {a(n)=local(L=vector(max(n,1), i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), sigma(m*k)*x^(m*k)/k)+x*O(x^n))))); if(n<1,0,L[n])}

%o (PARI) {a(n)=local(A=1+x+x*O(x^n),P=exp(sum(k=1,n,sigma(k)*x^k/k)+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(P, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

%Y Cf. A203320; A000041, A182818, A182819, A182820, A182821.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 31 2011