A385641 Partial sums of A097893.

1, 3, 8, 20, 51, 133, 356, 972, 2695, 7557, 21372, 60840, 174097, 500295, 1442720, 4172752, 12099411, 35161001, 102375400, 298586652, 872177273, 2551118623, 7471195500, 21904500500, 64286141881, 188844619563, 555216323396, 1633658183432, 4810340397375, 14173698242137

Offset

0,2

Comments

Second partial sums of the central trinomial coefficients (A002426).

Third partial sums of A025178 (sequence starting 1, 0, 2, 4, 12, 32, 90 .... with offset 0).

For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).

For p Pythagorean prime (A002144), a(p) - 3 == 0 (mod p).

Sequences with g.f. (1-x)^k / sqrt(1-2*x-3*x^2): this sequence (k=-2), A097893 (k=-1), A002426 (k=0), A025178 (k=1), A024997 (k=2), A026083 (k=3). - Mélika Tebni, Aug 25 2025

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1 / sqrt((1 + x)*(1 - 3*x))) / (1 - x)^2.

E.g.f.: exp(x)*(BesselI(0, 2*x) + 2*g(x) + Integral_{x=-oo..oo} g(x) dx) where g(x) = Integral_{x=-oo..oo} BesselI(0, 2*x) dx.

D-finite with recurrence n*a(n) = (4*n-1)*a(n-1) - (2*n+1)*a(n-2) - (4*n-5)*a(n-3) + 3*(n-1)*a(n-4).

a(0) = 1, a(1) = 3 and a(n) = a(n-2) - 1 + 2*A383527(n) for n >= 2.

a(n) = Sum_{k=0..n} binomial(n+1, k+1)*A128014(k).

a(n) = Sum_{k=0..n} (2*A247287(k) + k+1).

a(n) ~ 3^(n + 5/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 03 2025

Sum_{k=0..n} A295112(n-k)*a(k) + binomial(n+3, 3) = 0. - Mélika Tebni, Sep 03 2025

a(n) = A211278(n) + A247287(n). - Mélika Tebni, Sep 25 2025

From Mélika Tebni, Oct 03 2025: (Start)

a(n) = Sum_{k=0..n} A389359(n, k).

a(n) + a(n+1) = A389359(n+2, n+1). (End)

From Mélika Tebni, Oct 25 2025: (Start)

a(n) = Sum_{k=1..n+1} (2*k-2*n-1)*A132894(k).

a(n) = Sum_{k=0..n} (k-n+1)*A211288(k). (End)

Maple

a := series(exp(x)*(BesselI(0, 2*x) + 2*int(BesselI(0, 2*x), x) + int(int(BesselI(0, 2*x), x), x)), x = 0, 30): seq(n!*coeff(a, x, n), n = 0 .. 29);

Mathematica

Module[{x}, CoefficientList[Series[1/(Sqrt[(1 + x)*(1 - 3*x)]*(1 - x)^2), {x, 0, 30}], x]] (* Paolo Xausa, Nov 03 2025 *)

Prog

(Python)
from math import comb as C
def a(n):
    return sum(C(n+1, k+1)*C(2*(k//2), k//2) for k in range(n + 1))
print([a(n) for n in range(30)])
(PARI) a(n) = sum(k=0, n, sum(i=0, k, sum(j=0, i, binomial(i, i-j)*binomial(j, i-j)))); \\ Michel Marcus, Aug 06 2025

Crossrefs

Cf. A002144, A002145, A002426, A024997, A025178, A026083, A097893, A128014, A132894, A211278, A211288, A247287, A295112, A383527, A389359.

Sequence in context: A077997 A294407 A295346 * A027220 A393567 A305823

Adjacent sequences: A385638 A385639 A385640 * A385642 A385643 A385644

Keyword

nonn,easy

Author

Mélika Tebni, Aug 03 2025

Status

approved