A365690 G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^4 / (1 - x*A(x)).

1, 0, 1, 1, 5, 10, 38, 101, 353, 1070, 3659, 11843, 40505, 135873, 468104, 1604375, 5576315, 19386656, 67950717, 238676813, 842797959, 2983745508, 10603445402, 37777263153, 134985354179, 483438728094, 1735527037388, 6243193190117, 22503637842423

Offset

0,5

Links

Robert Israel, Table of n, a(n) for n = 0..1725

Formula

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

D-finite with recurrence: -3*(3*n + 2)*(3*n + 1)*(1 + n)*a(n) + (n + 2)*(167*n^2 + 615*n + 522)*a(1 + n) - 6*(19*n^3 + 365*n^2 + 1448*n + 1628)*a(n + 2) - (887*n^3 + 6774*n^2 + 14397*n + 5494)*a(n + 3) + 3*(9 + 2*n)*(287*n^2 + 2583*n + 5482)*a(n + 4) - (887*n^3 + 17175*n^2 + 108006*n + 222008)*a(n + 5) - 6*(19*n^3 + 148*n^2 - 505*n - 4310)*a(n + 6) + (n + 7)*(167*n^2 + 2391*n + 8514)*a(n + 7) - 3*(3*n + 25)*(n + 8)*(3*n + 26)*a(n + 8) = 0. - Robert Israel, May 18 2026

Maple

f:= gfun:-rectoproc({(-27*n^3 - 54*n^2 - 33*n - 6)*a(n) + (167*n^3 + 949*n^2 + 1752*n + 1044)*a(1 + n) + (-114*n^3 - 2190*n^2 - 8688*n - 9768)*a(n + 2) + (-887*n^3 - 6774*n^2 - 14397*n - 5494)*a(n + 3) + (1722*n^3 + 23247*n^2 + 102633*n + 148014)*a(n + 4) + (-887*n^3 - 17175*n^2 - 108006*n - 222008)*a(n + 5) + (-114*n^3 - 888*n^2 + 3030*n + 25860)*a(n + 6) + (167*n^3 + 3560*n^2 + 25251*n + 59598)*a(n + 7) + (-27*n^3 - 675*n^2 - 5622*n - 15600)*a(n + 8), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 5, a(5) = 10, a(6) = 38, a(7) = 101}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 18 2026

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n-k-1, n-2*k)*binomial(n+2*k+1, k)/(n+2*k+1));

Crossrefs

Cf. A004148, A005043, A025246, A046736, A365691.

Cf. A365692.

Sequence in context: A270219 A270276 A271604 * A277816 A359021 A355327

Adjacent sequences: A365687 A365688 A365689 * A365691 A365692 A365693

Keyword

nonn

Author

Seiichi Manyama, Sep 16 2023

Status

approved