cp's OEIS Frontend

A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.

%I A260361 #14 Aug 03 2023 07:42:54
%S A260361 2,4,2,10,2,12,16,16,2,50,24,24,58,28,72,154,2,36,302,40,142,270,332,
%T A260361 48,242,472,574,614,60,60,2282,64,2,1454,1362,2494,628,76,1940,3304,
%U A260361 3642,84,5266,88,662,13180,3544,96,994,6106,14292,13602,3434,108,8102,14854,16018,24778,7310,120,35684
%N A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.
%C A260361 Compare to the curious identities:
%C A260361 (1) Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.
%C A260361 (2) Sum_{n=-oo..+oo} (-x)^n * (1 + x^n)^n  =  0.
%F A260361 G.f.: Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n.
%F A260361 G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n.
%F A260361 G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n.
%F A260361 G.f.: Sum_{n>=1} x^(n^2-n) *((1 + x^n)^n + (1 - x^n)^n) / (1 - x^(2*n))^n.
%F A260361 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.
%F A260361 a(n) = 2*A260147(n).
%F A260361 a(2^n) = 2 for n > 0 (conjecture).
%F A260361 a(p) = 2*p+2 for primes p > 3 (conjecture).
%e A260361 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 +...
%e A260361 where A(x) = 1 + P(x) + N(x) with
%e A260361 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 +...
%e A260361 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 +...
%e A260361 Explicitly,
%e A260361 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 +...
%e A260361 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 +...
%t A260361 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 *)
%o A260361 (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)}
%o A260361 for(n=0, 60, print1(a(n), ", "))
%o A260361 (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)}
%o A260361 for(n=0, 60, print1(a(n), ", "))
%o A260361 (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)}
%o A260361 for(n=0, 60, print1(a(n), ", "))
%o A260361 (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)}
%o A260361 for(n=0, 60, print1(a(n), ", "))
%Y A260361 Cf. A260147, A217668, A260148, A363569, A363558, A363559, A363561.
%K A260361 nonn
%O A260361 0,1
%A A260361 _Paul D. Hanna_, Jul 23 2015