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 A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function. 1
 2, 4, 2, 10, 2, 12, 16, 16, 2, 50, 24, 24, 58, 28, 72, 154, 2, 36, 302, 40, 142, 270, 332, 48, 242, 472, 574, 614, 60, 60, 2282, 64, 2, 1454, 1362, 2494, 628, 76, 1940, 3304, 3642, 84, 5266, 88, 662, 13180, 3544, 96, 994, 6106, 14292, 13602, 3434, 108, 8102, 14854, 16018, 24778, 7310, 120, 35684 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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. LINKS Table of n, a(n) for n=0..60. FORMULA G.f.: Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n. G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n. G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n. G.f.: Sum_{n>=1} x^(n^2-n) *((1 + x^n)^n + (1 - x^n)^n) / (1 - x^(2*n))^n. G.f.: Sum_{n=-oo..+oo} x^n * ((1 + x^n)^(2*n) + (1 - x^n)^(2*n)) = Sum_{n>=0} a(n)*x^n. a(n) = 2*A260147(n). a(2^n) = 2 for n > 0 (conjecture). a(p) = 2*p+2 for primes p > 3 (conjecture). EXAMPLE G.f.: A(x) = 2 + 4*x^2 + 2*x^4 + 10*x^6 + 2*x^8 + 12*x^10 + 16*x^12 + 16*x^14 + 2*x^16 + 50*x^18 + 24*x^20 +... where 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; 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 + 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 + 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=1, sqrtint(2*n)+2, x^(k^2-k) *((1 + x^k)^k + (1 - x^k)^k) / (1 - x^(2*k) + O(x^(2*n+2)) )^k ); 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)) + O(x^(n+1)) ); polcoeff(A, n)} for(n=0, 60, print1(a(n), ", ")) CROSSREFS Cf. A260147, A217668, A260148, A363569, A363558, A363559, A363561. Sequence in context: A353108 A072866 A061393 * A055935 A086930 A235798 Adjacent sequences: A260358 A260359 A260360 * A260362 A260363 A260364 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 23 2015 STATUS approved

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Last modified September 27 20:41 EDT 2023. Contains 365714 sequences. (Running on oeis4.)