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A197707 G.f.: A(x) = 1 + Sum_{n>=1} x^(n^2) * ((1-x)^n + 1/(1-x)^n). 3

%I #15 Sep 03 2017 16:05:45

%S 1,2,0,1,3,1,5,5,6,9,8,18,19,26,33,41,52,60,87,99,132,166,209,261,323,

%T 398,481,604,716,893,1086,1331,1629,1991,2428,2952,3578,4314,5217,

%U 6229,7508,8967,10737,12838,15345,18334,21894,26127,31149,37093,44100

%N G.f.: A(x) = 1 + Sum_{n>=1} x^(n^2) * ((1-x)^n + 1/(1-x)^n).

%H Vaclav Kotesovec, <a href="/A197707/b197707.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: A(x) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^(2*n-1)*(1-x)) * (1 + x^(2*n-1)/(1-x)), due to the Jacobi triple product identity.

%e G.f.: A(x) = 1 + 2*x + x^3 + 3*x^4 + x^5 + 5*x^6 + 5*x^7 + 6*x^8 +...

%e where the g.f. A(x) may be expressed as the q-series:

%e A(x) = 1 + x*((1-x) + 1/(1-x)) + x^4*((1-x)^2 + 1/(1-x)^2) + x^9*((1-x)^3 + 1/(1-x)^3) + x^16*((1-x)^4 + 1/(1-x)^4) +...

%e and the Jacobi triple product:

%e A(x) = (1-x^2)*(1+x*(1-x))*(1+x/(1-x)) * (1-x^4)*(1+x^3*(1-x))*(1+x^3/(1-x)) * (1-x^6)*(1+x^5*(1-x))*(1+x^5/(1-x)) *...

%o (PARI) {a(n)=local(A); A=1+sum(m=1, sqrtint(n)+1, x^(m^2)*((1-x)^m+1/(1-x+x*O(x^n))^m)); polcoeff(A, n)}

%o (PARI) {a(n)=local(A); A=prod(m=1, n\2+1, (1-x^(2*m))*(1+x^(2*m-1)*(1-x))*(1+x^(2*m-1)/(1-x+x*O(x^n)))); polcoeff(A, n)}

%Y Cf. A190791.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 17 2011

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