A203320 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n*k) * x^(n*k)/k ) ).
1, 1, 2, 4, 9, 16, 35, 61, 124, 222, 427, 749, 1434, 2493, 4585, 8032, 14511, 25096, 44791, 77019, 135435, 232002, 402957, 685582, 1181399, 1998168, 3410288, 5741978, 9726821, 16286497, 27409625, 45672026, 76378731, 126706567, 210690588, 347954716, 575685559, 946756712
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 35*x^6 + 61*x^7 +...
G.f.: A(x) = exp( Sum_{n>=1} P_n(x^n) * x^n/n )
where P_n(x) = exp( Sum_{k>=1} sigma(n*k)*x^k/k ), which begin:
P_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...;
P_2(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...;
P_3(x) = 1 + 4*x + 14*x^2 + 39*x^3 + 101*x^4 + 238*x^5 + 533*x^6 +...;
P_4(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 + 3057*x^6 +...;
P_5(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + 2486*x^6 +...;
P_6(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 +...;
P_7(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 +...;
P_8(x) = 1 + 15*x + 128*x^2 + 815*x^3 + 4289*x^4 + 19663*x^5 +...;
...
Also, P_n(x^n) = Product_{k=0..n-1} P(u^k*x) where u = n-th root of unity:
P_1(x) = P(x), the partition function;
P_2(x^2) = P(x)*P(-x);
P_3(x^3) = P(x)*P(u*x)*P(u^2*x) where u = exp(2*Pi*I/3);
P_4(x^4) = P(x)*P(I*x)*P(I^2*x)*P(I^3*x) where I^2 = -1;
...
The logarithmic derivative of this sequence begins:
A203321 = [1,3,7,19,26,75,78,211,241,518,463,1447,1002,2558,...].
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..120
Programs
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PARI
{a(n)=local(L=vector(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))))); polcoeff(exp(x*Ser(vector(n+1, m, L[m]/m))), n)} -
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)))))); polcoeff(A, n)}