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 A326607 G.f.: Sum_{n>=0} (2*n+1) * x^n * (1 - x^n)^n. 1

%I

%S 1,3,2,7,-1,11,-3,15,-19,40,-34,23,-6,27,-76,141,-139,35,61,39,-270,

%T 358,-208,47,-35,106,-298,739,-874,59,725,63,-1415,1332,-526,596,-617,

%U 75,-664,2185,-2069,83,1519,87,-4696,5740,-988,95,-3571,204,3236,4863,-8995,107,1891,7701,-11578,6784,-1594,119,-941

%N G.f.: Sum_{n>=0} (2*n+1) * x^n * (1 - x^n)^n.

%C Compare the g.f. to the series: Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)) = Product_{n>=1} (1 - x^(2*n))^3.

%C Compare the g.f. to the series: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

%H Paul D. Hanna, <a href="/A326607/b326607.txt">Table of n, a(n) for n = 0..2550</a>

%F G.f.: Sum_{n>=0} (2*n+1) * x^n * (1 - x^n)^n.

%F G.f.: Sum_{n>=0} (-1)^n * (2*n+1 + x^(n+1)) * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).

%e G.f.: A(x) = 1 + 3*x + 2*x^2 + 7*x^3 - x^4 + 11*x^5 - 3*x^6 + 15*x^7 - 19*x^8 + 40*x^9 - 34*x^10 + 23*x^11 - 6*x^12 + 27*x^13 - 76*x^14 + 141*x^15 + ...

%e where A(x) is equal to the series given by

%e A(x) = 1 + 3*x*(1-x) + 5*x^2*(1-x^2)^2 + 7*x^3*(1-x^3)^3 + 9*x^4*(1-x^4)^4 + 11*x^5*(1-x^5)^5 + 13*x^6*(1-x^6)^6 + 15*x^7*(1-x^7)^7 + ...

%e Also,

%e A(x) = (1 + x)/(1 - x)^2 - (3 + x^2)*x^2/(1 - x^2)^3 + (5 + x^3)*x^6/(1 - x^3)^4 - (7 + x^4)*x^12/(1 - x^4)^5 + (9 + x^5)*x^20/(1 - x^5)^6 - (11 + x^6)*x^30/(1 - x^6)^7 + (13 + x^7)*x^42/(1 - x^7)^8 + ...

%o (PARI) /* By definition */

%o {a(n) = my(A = sum(m=0, n, (2*m + 1) * x^m * (1 - x^m + x*O(x^n))^m)); polcoeff(A, n)}

%o for(n=0, 60, print1(a(n), ", "))

%o (PARI) /* Accelerated series */

%o {a(n) = my(A = sum(m=0, sqrtint(n+1), (-1)^m * (2*m + 1 + x^(m+1))* x^(m*(m+1)) / (1 - x^(m+1) + x*O(x^n))^(m+2) )); polcoeff(A, n)}

%o for(n=0, 60, print1(a(n), ", "))

%Y Cf. A326605, A260180.

%K sign

%O 0,2

%A _Paul D. Hanna_, Oct 08 2019

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Last modified April 9 07:30 EDT 2020. Contains 333344 sequences. (Running on oeis4.)