A365729 G.f. satisfies A(x) = 1 + x^4*A(x)^3*(1 + x*A(x)).

1, 0, 0, 0, 1, 1, 0, 0, 3, 7, 4, 0, 12, 45, 55, 22, 55, 286, 546, 455, 413, 1820, 4760, 6120, 5304, 12597, 38760, 67830, 73587, 108262, 309925, 672980, 928763, 1171390, 2598310, 6270030, 10607025, 14033565, 24293880, 57256875, 112293597, 167416275, 255805056

Offset

0,9

Links

Robert Israel, Table of n, a(n) for n = 0..3707

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(n-k+1,k) / (n-k+1).

D-finite with recurrence: (243*n^3 + 972*n^2 + 1269*n + 540)*a(n) + (126*n^3 + 6768*n^2 + 27054*n + 27972)*a(n + 1) + (-47448*n^3 - 503136*n^2 - 1759356*n - 2030940)*a(n + 2) + (172785*n^3 + 2312955*n^2 + 10149540*n + 14625360)*a(n + 3) + (396666*n^3 + 6529950*n^2 + 35508144*n + 63832140)*a(n + 4) + (169272*n^3 + 3156408*n^2 + 19273716*n + 38482680)*a(n + 5) + (257077*n^3 + 6491586*n^2 + 54305633*n + 150475224)*a(n + 6) + (267765*n^3 + 7370040*n^2 + 67488855*n + 205535160)*a(n + 7) + (48840*n^3 + 1802760*n^2 + 20940600*n + 78035160)*a(n + 8) + (-10573*n^3 - 139905*n^2 + 416578*n + 7525800)*a(n + 9) + (-3216*n^3 - 105999*n^2 - 1170753*n - 4335780)*a(n + 10) + (-26004*n^3 - 856272*n^2 - 9322716*n - 33551568)*a(n + 11) + (-8132*n^3 - 301044*n^2 - 3696184*n - 15044160)*a(n + 12) + (-2025*n^3 - 83025*n^2 - 1132470*n - 5138760)*a(n + 13) + (216*n^3 + 9720*n^2 + 145704*n + 727560)*a(n + 14) = 0. - Robert Israel, Mar 02 2026

Maple

f:= gfun:-rectoproc({(243*n^3 + 972*n^2 + 1269*n + 540)*a(n) + (126*n^3 + 6768*n^2 + 27054*n + 27972)*a(n + 1) + (-47448*n^3 - 503136*n^2 - 1759356*n - 2030940)*a(n + 2) + (172785*n^3 + 2312955*n^2 + 10149540*n + 14625360)*a(n + 3) + (396666*n^3 + 6529950*n^2 + 35508144*n + 63832140)*a(n + 4) + (169272*n^3 + 3156408*n^2 + 19273716*n + 38482680)*a(n + 5) + (257077*n^3 + 6491586*n^2 + 54305633*n + 150475224)*a(n + 6) + (267765*n^3 + 7370040*n^2 + 67488855*n + 205535160)*a(n + 7) + (48840*n^3 + 1802760*n^2 + 20940600*n + 78035160)*a(n + 8) + (-10573*n^3 - 139905*n^2 + 416578*n + 7525800)*a(n + 9) + (-3216*n^3 - 105999*n^2 - 1170753*n - 4335780)*a(n + 10) + (-26004*n^3 - 856272*n^2 - 9322716*n - 33551568)*a(n + 11) + (-8132*n^3 - 301044*n^2 - 3696184*n - 15044160)*a(n + 12) + (-2025*n^3 - 83025*n^2 - 1132470*n - 5138760)*a(n + 13) + (216*n^3 + 9720*n^2 + 145704*n + 727560)*a(n + 14), a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1, a(5) = 1, a(6) = 0, a(7) = 0, a(8) = 3, a(9) = 7, a(10) = 4, a(11) = 0, a(12) = 12, a(13) = 45}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 02 2026

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(k, n-4*k)*binomial(n-k+1, k)/(n-k+1));

Crossrefs

Cf. A365727, A365728, A365730, A365731.

Cf. A365697.

Sequence in context: A050393 A110778 A200129 * A181912 A108297 A135928

Adjacent sequences: A365726 A365727 A365728 * A365730 A365731 A365732

Keyword

nonn

Author

Seiichi Manyama, Sep 17 2023

Status

approved