A344395 a(n) = binomial(4*n - 1, 2*n - 1)*hypergeom([-n, -n + 1/2], [2*n + 1], 4).

1, 5, 133, 4037, 129285, 4266830, 143567173, 4896136845, 168640510725, 5853000551090, 204368928058958, 7170955214476509, 252638095187722437, 8931025389858103602, 316640855103349347725, 11254413331736554364987, 400893874585938826203909, 14307778459379093347171266

Offset

0,2

Comments

Related to the Motzkin triangle A064189 counting certain lattice paths.

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Sum_{j=0..4*n-1} C(4*n-1, j)*(C(4*n-1-j, j+2*n-1) - C(4*n-1-j, j+2*n+1)) for n >= 1.

a(n) = A064189(4*n - 1, 2*n - 1) for n >= 1.

a(n) = A344394(4*n - 1) for n >= 1.

a(n) ~ sqrt(1014 + 156*sqrt(13)) * (13688 + 3640*sqrt(13))^n / (52 * sqrt(Pi*n) * 3^(6*n+1)). - Vaclav Kotesovec, Feb 18 2024

D-finite with recurrence +9*n*(6*n-1)*(3*n-1)*(3835115277622*n -6057563812695) *(2*n-1)*(3*n-2) *(6*n-5)*a(n) +2*(776430552534185648*n^7 -13254965233720706112*n^6 +77698256107321929944*n^5 -233839293644869788720*n^4 +406279253239920624227*n^3 -412808144693534857728*n^2 +228023561050132883751*n -52874097275943488160)*a(n-1) -108*(4*n-5)*(4*n-7) *(51631651831183544*n^5 -528937515408392660*n^4 +2125620894576233062*n^3 -4194554621940993427*n^2 +4055650255694760927*n -1531029729082241880)*a(n-2) +402408*(4*n-11)*(n-2) *(4*n-5)*(4*n-9)*(330342177838*n -391995025711)*(2*n-5) *(4*n-7)*a(n-3)=0. - R. J. Mathar, Mar 25 2024

Maple

alias(C=binomial):
a := n -> `if`(n = 0, 1, add(C(4*n - 1, j)*(C(4*n - 1 - j, j + 2*n - 1) - C(4*n - 1 - j, j + 2*n + 1)), j = 0..4*n-1)): seq(a(n), n = 0..17);

Mathematica

a[n_] := Binomial[4 n - 1, 2 n - 1] Hypergeometric2F1[-n, -n + 1/2, 2 n + 1, 4];
Table[a[n], {n, 0, 19}]

Crossrefs

Cf. A064189, A344394, A327871.

Sequence in context: A053517 A296191 A114474 * A280305 A085506 A307084

Adjacent sequences: A344392 A344393 A344394 * A344396 A344397 A344398

Keyword

nonn

Author

Peter Luschny, May 19 2021

Status

approved