A218153 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)) ).
1, 1, 2, 3, 6, 9, 16, 25, 42, 65, 105, 162, 256, 391, 605, 918, 1401, 2106, 3176, 4739, 7076, 10482, 15518, 22833, 33556, 49068, 71633, 104153, 151155, 218609, 315562, 454150, 652343, 934559, 1336328, 1906307, 2714409, 3856777, 5470236, 7743437, 10942743, 15435773
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 25*x^7 +... where log(A(x)) = x/1*((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)*...) + x^2/2*((1+x^2)*(1+x^4)*(1+x^6)*(1+x^8)*(1+x^10)*...) + x^3/3*((1+x^3)*(1+x^6)*(1+x^9)*(1+x^12)*(1+x^15)*...) + x^4/4*((1+x^4)*(1+x^8)*(1+x^12)*(1+x^16)*(1+x^20)*...) + x^5/5*((1+x^5)*(1+x^10)*(1+x^15)*(1+x^20)*(1+x^25)*...) +... Also, the g.f. is equal to the Euler transform of the distinct partitions A000009: A(x) = 1/((1-x)^1*(1-x^2)^1*(1-x^3)^1*(1-x^4)^2*(1-x^5)^2*(1-x^6)^3*(1-x^7)^4*(1-x^8)^5*(1-x^9)^6*(1-x^10)^8*(1-x^11)^10*...*(1-x^n)^A000009(n-1)*...).
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PartitionsQ[k-1], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 08 2016 *) -
PARI
{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*prod(k=1, n\m+1, 1+x^(m*k)+x*O(x^n)))), n)} for(n=0, 50, print1(a(n), ", "))
Formula
G.f.: Product_{n>=1} 1 / (1 - x^n)^A000009(n-1), where A000009(n) equals the number of distinct partitions of n. - Paul D. Hanna, Nov 16 2012
Comments