A391811 a(n) equals half the sum of the squares of the terms in row n of irregular triangle A391810.

1, 1, 4, 13, 53, 199, 674, 2313, 8731, 34453, 135151, 516056, 1929917, 7191767, 27150041, 104424657, 407094655, 1594363303, 6233521880, 24277356505, 94261319494, 365585759896, 1419238266986, 5523689999096, 21573875791838, 84572021687714, 332542322089642, 1309933674885610

Offset

1,3

Comments

Row n of irregular triangle A391810 consists of only the nonpositive powers of x in the doubly infinite series S(n) = Sum_{k=-oo..+oo} x^k * (1 - x^k)^(n+k) for n >= 1, with S(0) = 0. The index k in A391810(n,k) goes from k = 0 to k = floor((n+1)/2)^2.

The row sums of A391810 appear to equal zero for rows n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..150

Formula

a(n) = (1/2) * Sum_{k=0..floor((n+1)/2)^2} A391810(n,k)^2.

Prog

(PARI) {R(n) = my(N=((n+1)\2)^2); x^N*( sum(k=-N-2, N+2, x^k * (1 - x^k +x*O(x^N))^(n+k) ) + O(x) )}
{A391810(n, k) = polcoef(R(n), ((n+1)\2)^2 - k)}
for(n=1, 30, print1( (1/2)*sum(k=0, ((n+1)\2)^2, A391810(n, k)^2), ", "))

Crossrefs

Cf. A391810.

Sequence in context: A129147 A151476 A140454 * A149465 A149466 A369226

Adjacent sequences: A391808 A391809 A391810 * A391812 A391813 A391814

Keyword

nonn

Author

Paul D. Hanna, Dec 21 2025

Status

approved