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

Author

Mélika Tebni, Aug 03 2025

Keywords

• nonn
• easy
• changed

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

Crossrefs

Cf. A002144, A002145, A002426, A024997, A025178, A026083, A097893, A128014, A247287, A383527.

Programs

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);

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

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)])

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