A391462 Expansion of g/(2 - g^2), where g = 1+x*g^3 is the g.f. of A001764.

1, 3, 16, 95, 590, 3755, 24266, 158448, 1042346, 6895491, 45814172, 305442703, 2042080348, 13684126812, 91875463080, 617863215368, 4160959900522, 28055667406147, 189367714620976, 1279362095322717, 8650402250562118, 58532189534911523, 396310088499317090

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1193

Formula

a(n) = (1/n) * Sum_{k=1..n} k * (Pell(k) + Pell(k+1)) * binomial(3*n,n-k) for n > 0.

D-finite with recurrence: (5184*n^2 + 5184*n + 1152)*a(n) + (4416*n^2 + 34368*n + 36288)*a(n + 1) + (-13728*n^2 - 70032*n - 85824)*a(n + 2) + (6672*n^2 + 42312*n + 66936)*a(n + 3) + (-1325*n^2 - 10499*n - 20850)*a(n + 4) + (119*n^2 + 1148*n + 2772)*a(n + 5) + (-4*n^2 - 46*n - 132)*a(n + 6) = 0. - Robert Israel, Feb 12 2026

Maple

f:= gfun:-rectoproc({(5184*n^2 + 5184*n + 1152)*a(n) + (4416*n^2 + 34368*n + 36288)*a(n + 1) + (-13728*n^2 - 70032*n - 85824)*a(n + 2) + (6672*n^2 + 42312*n + 66936)*a(n + 3) + (-1325*n^2 - 10499*n - 20850)*a(n + 4) + (119*n^2 + 1148*n + 2772)*a(n + 5) + (-4*n^2 - 46*n - 132)*a(n + 6) = 0, a(0) = 1, a(1) = 3, a(2) = 16, a(3) = 95, a(4) = 590, a(5) = 3755}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 12 2026

Prog

(PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n) = if(n==0, 1, sum(k=1, n, k*(pell(k)+pell(k+1))*binomial(3*n, n-k))/n);

Crossrefs

Cf. A391459, A391465.

Cf. A000129.

Sequence in context: A213229 A323968 A074555 * A137644 A114174 A181067

Adjacent sequences: A391459 A391460 A391461 * A391463 A391464 A391465

Keyword

nonn

Author

Seiichi Manyama, Dec 10 2025

Status

approved