%I #19 Aug 11 2019 19:28:54

%S 1,3,0,2,9,-12,8,12,-10,15,-36,46,-18,20,12,-82,101,-162,302,-168,

%T -100,92,32,40,-244,351,-452,1052,-1528,1144,-394,-494,948,-438,370,

%U -1474,2805,-2860,2560,-4762,6554,-4104,926,480,-1820,3074,-2546,1072,-2518,9745,-17810,21300,-28982,37560,-26380,6162,686,2,2364,-12342,30356,-39584,19448,7562,9491,-63824,99128,-116668

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

%H Paul D. Hanna, <a href="/A326602/b326602.txt">Table of n, a(n) for n = 0..4100</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n), where

%F (1) A(x) = Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n / (1 + x^(2*n+1))^(n+1),

%F (2) A(x) = Sum_{n>=0} (x^(2*n-1) - 1)^n * x^n / (1 - x^(2*n+1))^(n+1),

%F (3) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (x^(2*n-1) - x^(2*k))^(n-k),

%F (4) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (x^(2*n-1) + x^(2*k))^(n-k) * (-1)^k.

%F a(n) is odd iff n is square (conjecture).

%e G.f.: A(x) = 1 + 3*x^2 + 2*x^6 + 9*x^8 - 12*x^10 + 8*x^12 + 12*x^14 - 10*x^16 + 15*x^18 - 36*x^20 + 46*x^22 - 18*x^24 + 20*x^26 + 12*x^28 - 82*x^30 + ...

%e such that

%e A(x) = 1/(1 + x) + (1 + x)*x/(1 + x^3)^2 + (1 + x^3)^2*x^2/(1 + x^5)^3 + (1 + x^5)^3*x^3/(1 + x^7)^4 + (1 + x^7)^4*x^4/(1 + x^9)^5 + (1 + x^9)^5*x^5/(1 + x^11)^6 + (1 + x^11)^6*x^6/(1 + x^13)^7 + (1 + x^13)^7*x^7/(1 + x^15)^8 + ...

%e also

%e A(x) = 1/(1 - x) - (1 - x)*x/(1 - x^3)^2 + (1 - x^3)^2*x^2/(1 - x^5)^3 - (1 - x^5)^3*x^3/(1 - x^7)^4 + (1 - x^7)^4*x^4/(1 - x^9)^5 - (1 - x^9)^5*x^5/(1 - x^11)^6 + (1 - x^11)^6*x^6/(1 - x^13)^7 - (1 - x^13)^7*x^7/(1 - x^15)^8 + ...

%e AS A TRIANGLE.

%e This sequence may be written as a triangle like so

%e 1,

%e 3, 0, 2,

%e 9, -12, 8, 12, -10,

%e 15, -36, 46, -18, 20, 12, -82,

%e 101, -162, 302, -168, -100, 92, 32, 40, -244,

%e 351, -452, 1052, -1528, 1144, -394, -494, 948, -438, 370, -1474,

%e 2805, -2860, 2560, -4762, 6554, -4104, 926, 480, -1820, 3074, -2546, 1072, -2518,

%e 9745, -17810, 21300, -28982, 37560, -26380, 6162, 686, 2, 2364, -12342, 30356, -39584, 19448, 7562,

%e 9491, -63824, 99128, -116668, 167616, -212884, 156266, -32564, -35108, 21732, 3952, -4058, -19496, 89988, -200198, 251662, -198964, ...

%e in which the terms a(n^2) form the leftmost column.

%e The odd terms seem to occur only at a(n^2) and begin:

%e [1, 3, 9, 15, 101, 351, 2805, 9745, 9491, 138675, 776675, 7430517, 43105515, ...].

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

%o polcoeff(A,2*n)}

%o for(n=0,100,print1(a(n),", "))

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

%o polcoeff(A,2*n)}

%o for(n=0,100,print1(a(n),", "))

%Y Cf. A326603, A323557.

%K sign

%O 0,2

%A _Paul D. Hanna_, Aug 08 2019