A248586 -id:A248586 - cp's OEIS frontend

A079646 A congruence property: a(n) = (A248586(p)-5)/(4*p) where p is the n-th prime.

5, 43, 4619, 688390, 23517593780, 4879569381720, 239177513351286095, 55404294334042254028, 3163613989757561167486439, 48119240966975050455116308972625, 12175395259244110756232625177237480, 206515669664553406211418087534393094881690

Offset: 1

Benoit Cloitre, Feb 16 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..141

Cf. A248586.

f[n_] := Sum[Binomial[n, k] * Binomial[2*k, k]^2, {k, 0, n}]; Table[(f[p]-5)/(4*p), {p, Prime[Range[12]]}] (* Amiram Eldar, Apr 20 2025 *)

Name corrected by Amiram Eldar, Apr 20 2025

A276536 Binomial sums of the cubes of the central binomial coefficients.

Original entry on oeis.org

1, 9, 233, 8673, 376329, 17800209, 890215361, 46294813497, 2478150328777, 135642353562321, 7556884938829233, 427106589765940137, 24429206859151618209, 1411391470651692285609, 82245902444586364980057, 4828398428680134702936273

Offset: 0

Emanuele Munarini, Nov 16 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..554
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind
The Wolfram Functions Site, Complete Elliptic Integrals, 2016.

Cf. Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^m: A026375 (m=1), A248586 (m=2), this sequence (m=3).

[&+[Binomial(n, k)*Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 30 2016

Table[Sum[Binomial[n, k]Binomial[2k, k]^3, {k, 0, n}], {n, 0, 100}]

makelist(sum(binomial(n,k)*binomial(2*k,k)^3,k,0,n),n,0,12);

a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^3.
Recurrence: (n^3 + 12n^2 + 48n + 64) * a(n+4) - (68n^3 + 714n^2 + 2500n + 2919) * a(n+3) + (198n^3 + 1782n^2 + 5363n + 5397) * a(n+2) - 98 * (2n^3 + 15n^2 + 37n + 30) * a(n+1) + 65 * (n^3 + 6n^2 + 11n + 6) * a(n) = 0.
G.f.: (4/Pi^2) * K(1/2 - 1/2 * sqrt((1-65*t)/(1-t)))^2 / (1-t), where K(x) is complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).
a(n) ~ 65^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 16 2016
a(n) = 4F3(1/2,1/2,1/2,-n; 1,1,1; -64). - Ilya Gutkovskiy, Nov 25 2016

A278934 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)binomial(2*k,k)^2.

Original entry on oeis.org

1, 3, 29, 303, 3501, 42663, 538769, 6977547, 92078989, 1232902023, 16700233689, 228356672547, 3147087003201, 43659275921667, 609117615688149, 8539863624592023, 120242239301247309, 1699411957967345127, 24098616839012623769, 342754384909199620803

Offset: 0

Seiichi Manyama, Dec 02 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..853

Cf. A248586.

Cf. Sum_{k = 0..n} (-1)^(n-k)*binomial(n, k)*binomial(2*k, k)^m: A002426 (m=1), this sequence (m=2), A276537 (m=3).

Table[Sum[(-1)^(n-k)*Binomial[n,k]*Binomial[2*k,k]^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 02 2016 *)
Table[(-1)^n*HypergeometricPFQ[{1/2, 1/2, -n}, {1, 1}, 16], {n, 0, 20}] (* Vaclav Kotesovec, Dec 02 2016 *)

Recurrence: n^2*a(n) = (13*n^2 - 13*n + 3)*a(n-1) + 29*(n-1)^2*a(n-2) + 15*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Dec 02 2016

a(n) ~ 15^(n+1) / (16*Pi*n). - Vaclav Kotesovec, Dec 02 2016

cp's OEIS Frontend

A079646 A congruence property: a(n) = (A248586(p)-5)/(4*p) where p is the n-th prime.

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A276536 Binomial sums of the cubes of the central binomial coefficients.

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A278934 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)binomial(2*k,k)^2.

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cp's OEIS Frontend

A079646 A congruence property: a(n) = (A248586(p)-5)/(4*p) where p is the n-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A276536 Binomial sums of the cubes of the central binomial coefficients.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

Formula

A278934 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*binomial(2*k,k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A278934 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)binomial(2*k,k)^2.