A364373 -id:A364373 - cp's OEIS frontend

A364371 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^2).

1, 0, -1, 2, -2, -1, 9, -20, 20, 24, -150, 327, -293, -599, 3097, -6452, 4854, 15878, -71252, 140112, -81328, -437346, 1746254, -3214989, 1223971, 12345295, -44552833, 76242173, -11292089, -354175849, 1167638037, -1842585992, -233903034, 10273377388, -31169512310

Offset: 0

Seiichi Manyama, Jul 20 2023

Cf. A364372, A364373, A364375.

Cf. A000108, A073157.

A364371 := proc(n)
    add((-1)^k* binomial(2*k+1,k) * binomial(2*k+1,n-k)/(2*k+1),k=0..n) ;
end proc:
seq(A364371(n),n=0..70); # R. J. Mathar, Jul 25 2023

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

G.f.: A(x) = 2*(1 + x) / (1 + sqrt(1+4*x*(1 + x)^2)).
a(n) = Sum_{k=0..n} (-1)^k * binomial(2*k+1,k) * binomial(2*k+1,n-k) / (2*k+1).
D-finite with recurrence (n+1)*a(n) +(5*n-1)*a(n-1) +6*(2*n-3)*a(n-2) +6*(2*n-5)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 24 2024: (Start)
A(x) = (1 + x)*c(-x*(1+x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
(1/x) * series_reversion(x/A(x)) = 1 - x^2 + 2*x^3 - 11*x^5 + 28*x^6 + ..., the g.f. of A364375. (End)

A364372 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^3).

Original entry on oeis.org

1, 0, -1, 3, -6, 6, 15, -107, 349, -672, 39, 5835, -27654, 75765, -95799, -279129, 2297970, -8377854, 17663640, -996624, -177445221, 888491025, -2551959604, 3337931168, 10407149226, -87719805853, 328682535695, -708428979213, 15252552804, 7616368090377, -38693979668535

Offset: 0

Seiichi Manyama, Jul 20 2023

Cf. A364371, A364373, A364376.

Cf. A364336.

A364372 := proc(n)
    add( (-1)^k*binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
end proc:
seq(A364372(n),n=0..80); # R. J. Mathar, Jul 25 2023

a(n) = sum(k=0, n, (-1)^k*binomial(3*k+1, k)*binomial(3*k+1, n-k)/(3*k+1));

a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).
D-finite with recurrence 2*n*(2*n+1)*a(n) +(51*n-26)*(n-1)*a(n-1) +(279*n^2 -931*n +766)*a(n-2) +2*(413*n^2 -2127*n +2728)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 25 2024: (Start)
Fifth-order recurrence: 2*(n-2)*(n-1)*n*(2*n+1)*a(n) + (n-2)*(n-1)*(31*n^2-27*n+6)*a(n-1) + 3*(n-2)*(3*n-5)*(12*n^2-19*n+2)*a(n-2) + 3*(3*n-8)*(18*n^3-69*n^2+63*n-2)*a(n-3) + 3*n*(3*n-11)*(12*n^2-49*n+42)*a(n-4) + 3*n*(n-1)*(3*n-10)*(3*n-14)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 3 and a(4) = -6.
The g.f. A(x) satisfies (1/x) * series_reversion(x/A(x)) = 1 - x^2 + 3*x^3 - 4*x^4 - 9*x^5 + 73*x^6 - ..., the g.f. of A364376. (End)

cp's OEIS Frontend

A364371 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^2).

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