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 A260147 G.f.: (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function. 10
 1, 2, 1, 5, 1, 6, 8, 8, 1, 25, 12, 12, 29, 14, 36, 77, 1, 18, 151, 20, 71, 135, 166, 24, 121, 236, 287, 307, 30, 30, 1141, 32, 1, 727, 681, 1247, 314, 38, 970, 1652, 1821, 42, 2633, 44, 331, 6590, 1772, 48, 497, 3053, 7146, 6801, 1717, 54, 4051, 7427, 8009, 12389, 3655, 60, 17842, 62, 4496, 42841, 1, 15731, 6470, 68, 19449, 34754, 65781 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Compare to the curious identities: (1) Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0. (2) Sum_{n=-oo..+oo} (-x)^n * (1 + x^n)^n  =  0. Given G(x,q) = Sum_{n=-oo..+oo} (1 + q^n)^n * q^n * x^n, then [x^0] G(x,q)^2 = theta_3(q) = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 +... LINKS Paul D. Hanna, Table of n, a(n) for n = 0..2050 FORMULA The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: (1) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^n)^n. (2) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n. (3) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n. (4) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n. (5) A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^n)^(2*n). (6) A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^n)^(2*n). (7) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - x^n)^(2*n). (8) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + x^n)^(2*n). a(2^n) = 1 for n > 0 (conjecture). a(p) = p+1 for odd primes p > 3 (conjecture). From Peter Bala, Jan 23 2021: (Start) The following are conjectural: A(x^2) = Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1) )^(2*n+1). Equivalently: A(x^2) = Sum_{n = -oo..+oo} x^(4*n^2 + 2*n)/(1 + x^(2*n+1))^(2*n+1). a(2*n+1) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(4*n+2) More generally, for k = 1,2,3,..., a((2^k)*(2*n + 1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(2^(k+1)*(2*n+1)). a(2*n+1) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 + x^n)^(2*n) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 - x^n)^(2*n). More generally, for k = 1,2,3,..., a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 + x^n)^(2^(k+1)*n) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 - x^n)^(2^(k+1)*n). a(4*n+2) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 + x^n)^(2*n) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 - x^n)^(2*n). a(n) = [x^(2*n)] Sum_{n = -oo..+oo} (-1)^n*x^(2*n+1)*(1 + (-1)^n* x^(2*n+1) )^(2*n+1). For k = 1,2,3,..., a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + (-1)^n* x^(2*n+1) )^(2^(k+1)*(2*n+1)). (End) EXAMPLE G.f.: A(x) = 1 + 2*x^2 + x^4 + 5*x^6 + x^8 + 6*x^10 + 8*x^12 + 8*x^14 + x^16 + 25*x^18 + 12*x^20 +... where 2*A(x) = 1 + P(x) + N(x) with P(x) = x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5 +... N(x) = 1/(1+x) + x^2/(1+x^2)^2 + x^6/(1+x^3)^3 + x^12/(1+x^4)^4 + x^20/(1+x^5)^5 +... Explicitly, P(x) = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 + 4*x^9 + 6*x^10 + x^11 + 14*x^12 + x^13 + 8*x^14 + 11*x^15 + 13*x^16 + x^17 + 25*x^18 + x^19 + 22*x^20 + 22*x^21 + 12*x^22 + x^23 + 61*x^24 + 6*x^25 +...+ A217668(n)*x^n +... N(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 - x^11 + 2*x^12 - x^13 + 8*x^14 - 11*x^15 - 11*x^16 - x^17 + 25*x^18 - x^19 + 2*x^20 - 22*x^21 + 12*x^22 - x^23 - 3*x^24 - 6*x^25 +...+ A260148(n)*x^n +... MATHEMATICA terms = 100; max = 2 terms; 1/2 Sum[x^n*(1 + x^n)^n, {n, -max, max}] + O[x]^max // CoefficientList[#, x^2]& (* Jean-François Alcover, May 16 2017 *) PROG (PARI) {a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^k*(1+x^k)^k/2 + O(x^(2*n+2)) ); polcoeff(A, 2*n)} for(n=0, 60, print1(a(n), ", ")) (PARI) {a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^(k^2-k) / (1 + x^k)^k /2  + O(x^(2*n+2)) ); polcoeff(A, 2*n)} for(n=0, 60, print1(a(n), ", ")) (PARI) {a(n) = local(A=1); A = sum(k=-sqrtint(n)-1, n+1, x^k*((1+x^k)^(2*k) + (1-x^k)^(2*k))/2 + O(x^(n+1)) ); polcoeff(A, n)} for(n=0, 60, print1(a(n), ", ")) (PARI) {a(n) = local(A=1); A = sum(k=-n-1, n+1, x^k*(1+x^k)^(2*k) + O(x^(n+1)) ); polcoeff(A, n)} for(n=0, 60, print1(a(n), ", ")) (PARI) {a(n) = local(A=1); A = sum(k=-n-1, n+1, x^(2*k^2-k)/(1-x^k + O(x^(n+1)))^(2*k)  ); polcoeff(A, n)} for(n=0, 60, print1(a(n), ", ")) CROSSREFS Cf. A260116, A260148, A217668, A260180, A260361. Cf. A261605. Sequence in context: A249548 A014650 A014648 * A263454 A036073 A124227 Adjacent sequences:  A260144 A260145 A260146 * A260148 A260149 A260150 KEYWORD nonn,easy AUTHOR Paul D. Hanna, Jul 17 2015 STATUS approved

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Last modified December 6 20:22 EST 2021. Contains 349567 sequences. (Running on oeis4.)