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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A361710 a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n,k)*binomial(n-1,k)^2.

Original entry on oeis.org

0, 1, -1, -8, 15, 126, -280, -2400, 5775, 50050, -126126, -1100736, 2858856, 25069968, -66512160, -585307008, 1577585295, 13919870250, -37978905250, -335813478000, 925166131890, 8194328596740, -22754499243840, -201822515032320, 564121960420200, 5009403008531376
Offset: 0

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Author

Peter Bala, Mar 21 2023

Keywords

Comments

Conjecture: the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and positive integers n and k.
Compare with A005258(n-1) = Sum_{k = 0..n-1} (-1)^k*binomial(-n,k)*binomial(n-1,k)^2.

Links

Crossrefs

Cf. A005258, A006480, A245086, A361711, A361716.

Programs

  • Maple
    seq( add((-1)^k*binomial(n,k)*binomial(n-1,k)^2, k = 0..n-1), n = 0..25);
  • PARI
    a(n) = sum(k = 0, n-1, (-1)^k*binomial(n,k)*binomial(n-1,k)^2); \\ Michel Marcus, Mar 26 2023

Formula

a(n) = (1/n^2) * Sum_{k = 0..n} (-1)^(n+k) * k^2 * binomial(n,k)^3 for n >= 1.
a(n) = (1/(3*n)) * Sum_{k = 0..n} (-1)^(n+k+1) * (n - 3*k) * binomial(n,k)^3 for n >= 1.
a(2*n) = (-1)^n * (1/6) * (3*n)!/n!^3 for n >= 1; a(2*n+1) = (-1)^n * (3*n+1)/(2*n+1) * (3*n)!/n!^3.
a(2*n) = (1/3)*A361716(2*n); a(2*n+1) = A361711(2*n+1) = A361716(2*n+1).
a(2*n) = (1/6)*A245086(2*n) = (1/6)*(-1)^n*A006480(n) for n >= 1.
a(n) = hypergeom([-n, 1 - n, 1 - n], [1, 1], 1);
P-recursive: n^2*(n - 1)*(6*n^2 - 16*n + 11)*a(n) = - 6*(n - 1)*(3*n^2 - 6*n + 2)*a(n-1) - (3*n - 4)*(3*n - 5)*(3*n - 6)*(6*n^2 - 4*n + 1)*a(n-2) with a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n,k)*binomial(n+k-1,k)*binomial(2*n-k-1,n). - Peter Bala, Jul 01 2023

A382848 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * binomial(n+k,k).

Original entry on oeis.org

1, 1, -5, -35, -29, 751, 3991, -4115, -137885, -495269, 2114245, 25786795, 50109775, -627370925, -4643568305, -495798035, 157753390435, 768269873875, -1851203127335, -35924154988865, -107001450483779, 763444753890721, 7510024190977105, 8899910747771995
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

Comments

Diagonal of the rational function 1 / (1 + x + x*y + y*z + x*z + x*y*z).

Crossrefs

Cf. A005258, A026641, A126869, A245086, A382405, A382849.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[n + k, k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, -1], {n, 0, 23}]
Table[SeriesCoefficient[1/(1 + x + x y + y z + x z + x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 23}]

Formula

(59*n-94)*n^2*a(n) = 5*(59*n^3-153*n^2+117*n-30)*a(n-1) - (2301*n^3-8268*n^2+9257*n-3050)*a(n-2) - 2*(59*n-35)*(n-2)^2*a(n-3) with a(0) = 1, a(1) = 1 and a(2) = -5. - Peter Bala, May 24 2025
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