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A218153 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)) ).

%I A218153 #28 Apr 08 2016 08:58:37
%S A218153 1,1,2,3,6,9,16,25,42,65,105,162,256,391,605,918,1401,2106,3176,4739,
%T A218153 7076,10482,15518,22833,33556,49068,71633,104153,151155,218609,315562,
%U A218153 454150,652343,934559,1336328,1906307,2714409,3856777,5470236,7743437,10942743,15435773
%N A218153 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)) ).
%C A218153 Compare to the g.f. of A001383:
%C A218153 1 + x*exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)) ).
%H A218153 Paul D. Hanna, <a href="/A218153/b218153.txt">Table of n, a(n) for n = 0..1000</a>
%F A218153 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
%e A218153 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 25*x^7 +...
%e A218153 where
%e A218153 log(A(x)) = x/1*((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)*...) +
%e A218153 x^2/2*((1+x^2)*(1+x^4)*(1+x^6)*(1+x^8)*(1+x^10)*...) +
%e A218153 x^3/3*((1+x^3)*(1+x^6)*(1+x^9)*(1+x^12)*(1+x^15)*...) +
%e A218153 x^4/4*((1+x^4)*(1+x^8)*(1+x^12)*(1+x^16)*(1+x^20)*...) +
%e A218153 x^5/5*((1+x^5)*(1+x^10)*(1+x^15)*(1+x^20)*(1+x^25)*...) +...
%e A218153 Also, the g.f. is equal to the Euler transform of the distinct partitions A000009:
%e A218153 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)*...).
%t A218153 nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PartitionsQ[k-1], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 08 2016 *)
%o A218153 (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)}
%o A218153 for(n=0, 50, print1(a(n), ", "))
%Y A218153 Cf. A218552, A218576, A000009, A001383, A089259.
%K A218153 nonn
%O A218153 0,3
%A A218153 _Paul D. Hanna_, Nov 01 2012