A326602 G.f.: Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n / (1 + x^(2*n+1))^(n+1), an even function.

1, 3, 0, 2, 9, -12, 8, 12, -10, 15, -36, 46, -18, 20, 12, -82, 101, -162, 302, -168, -100, 92, 32, 40, -244, 351, -452, 1052, -1528, 1144, -394, -494, 948, -438, 370, -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

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..4100

Formula

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

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

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

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

(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.

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

Example

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 + ...
such that
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 + ...
also
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 + ...
AS A TRIANGLE.
This sequence may be written as a triangle like so
1,
3, 0, 2,
9, -12, 8, 12, -10,
15, -36, 46, -18, 20, 12, -82,
101, -162, 302, -168, -100, 92, 32, 40, -244,
351, -452, 1052, -1528, 1144, -394, -494, 948, -438, 370, -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, 167616, -212884, 156266, -32564, -35108, 21732, 3952, -4058, -19496, 89988, -200198, 251662, -198964, ...
in which the terms a(n^2) form the leftmost column.
The odd terms seem to occur only at a(n^2) and begin:
[1, 3, 9, 15, 101, 351, 2805, 9745, 9491, 138675, 776675, 7430517, 43105515, ...].

Prog

(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) ));
polcoeff(A, 2*n)}
for(n=0, 100, print1(a(n), ", "))
(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) ));
polcoeff(A, 2*n)}
for(n=0, 100, print1(a(n), ", "))

Crossrefs

Cf. A326603, A323557.

Sequence in context: A231101 A193084 A126598 * A256548 A239098 A319501

Adjacent sequences: A326599 A326600 A326601 * A326603 A326604 A326605

Keyword

sign

Author

Paul D. Hanna, Aug 08 2019

Status

approved