cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A344396 a(n) = binomial(2*n + 1, n)*hypergeom([-(n + 1)/2, -n/2], [n + 2], 4).

Original entry on oeis.org

1, 5, 25, 133, 726, 4037, 22737, 129285, 740554, 4266830, 24701425, 143567173, 837212650, 4896136845, 28703894775, 168640510725, 992671051482, 5853000551090, 34562387229046, 204368928058958, 1209916827501876, 7170955214476509, 42543879586512435, 252638095187722437
Offset: 0

Views

Author

Peter Luschny, May 19 2021

Keywords

Comments

Related to the Motzkin triangle A064189 counting certain lattice paths.

Crossrefs

Cf. A027908, A064189, A026300, A344394, A327871.

Programs

  • Maple
    alias(C=binomial):
a := n -> add(C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)), j = 0..2*n+1): seq(a(n), n=0..23);
  • Mathematica
    a[n_] := Binomial[2 n + 1, n] Hypergeometric2F1[-(n + 1)/2, -n/2, n + 2, 4];
Table[a[n], {n, 0, 23}]

Formula

a(n) = Sum_{j=0..2*n+1} C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)).
a(n) = A064189(2*n+1, n).
a(n) = A026300(2*n+1, n+1).
a(n) ~ sqrt((5242 + 18674/sqrt(13))/2187) * ((70 + 26*sqrt(13))/27)^n / sqrt(Pi*n). - Vaclav Kotesovec, May 19 2021
From Peter Bala, Aug 03 2023: (Start)
P-recursive: 3*(13*n - 4)*(3*n + 2)*(3*n + 1)*(n + 1)*a(n) = 2*(2*n + 1)*(455*n^3 + 315*n^2 - 44*n - 24)*a(n-1) + 36*(13*n + 9)*(2*n + 1)*(2*n - 1)*n*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = (1/2)*A027908(n+1). (End)

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

Original entry on oeis.org

1, 5, 133, 4037, 129285, 4266830, 143567173, 4896136845, 168640510725, 5853000551090, 204368928058958, 7170955214476509, 252638095187722437, 8931025389858103602, 316640855103349347725, 11254413331736554364987, 400893874585938826203909, 14307778459379093347171266
Offset: 0

Views

Author

Peter Luschny, May 19 2021

Keywords

Comments

Related to the Motzkin triangle A064189 counting certain lattice paths.

Crossrefs

Cf. A064189, A344394, A327871.

Programs

  • 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}]

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
Showing 1-2 of 2 results.

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