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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A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k).

Original entry on oeis.org

1, 2, 11, 62, 367, 2232, 13820, 86662, 548591, 3498146, 22436251, 144583496, 935394436, 6071718512, 39523955552, 257913792342, 1686627623151, 11050540084902, 72522925038257, 476669316338542, 3137209052543927, 20672732229560032, 136374124374593072, 900541325129687272
Offset: 0

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Author

Paul D. Hanna, Dec 27 2010

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 +...
A(x)^(1/2) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...+ A183161(n)*x^n +...
Given triangle A085478(n,k) = C(n+k,n-k), which begins:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  5,  1;
  1, 10, 15,  7, 1;
  1, 15, 35, 28, 9, 1; ...
ILLUSTRATE formula a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k):
a(2) = 11 = 1*1 + 3*3 + 1*1;
a(3) = 62 = 1*1 + 6*5 + 5*6 + 1*1;
a(4) = 367 = 1*1 + 10*7 + 15*15 + 7*10 + 1*1;
a(5) = 2232 = 1*1 + 15*9 + 35*28 + 28*35 + 9*15 + 1*1; ...

Links

Crossrefs

Cf. A001764, A085478, A183161, A199033, A218622.
Cf. A026641, A147855, A371753.
Cf. A005809, A006256, A160906, A226751.
Cf. A171822.

Programs

  • Magma
    [(&+[Binomial(n+k, 2*k)*Binomial(2*n-k, k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 22 2021
  • Mathematica
    Table[Sum[Binomial[n+k,n-k]Binomial[2n-k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)
Table[HypergeometricPFQ[{-n, -n, 1/2 -n, n+1}, {1/2, 1, -2*n}, 1], {n, 0, 25}] (* G. C. Greubel, Feb 22 2021 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(2*n-k,k))}
  • PARI
    {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1-2*x*G^2-3*x^2*G^4), n)} \\ Paul D. Hanna, Nov 03 2012
  • PARI
    {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1+3*x*G-5*x*G^2), n)} \\ Paul D. Hanna, Jun 16 2013
for(n=0, 30, print1(a(n), ", "))
  • Sage
    a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],2)
[simplify(a(n)) for n in range(26)] # Peter Luschny, May 19 2015

Formula

a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k).
Self-convolution of A183161 (an integer sequence):
a(n) = Sum_{k=0..n} A183161(k)*A183161(n-k).
a(n) = Sum_{k=0..n} binomial(2*n+k,k) * cos((n+k)*Pi). - Arkadiusz Wesolowski, Apr 02 2012
Recurrence: 320*n*(2*n-1)*a(n) = 8*(346*n^2 + 79*n - 327)*a(n-1) + 6*(1688*n^2-6241*n+5981)*a(n-2) + 261*(3*n-7)*(3*n-5)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3^(3*n+3/2)/(2^(2*n+3)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
...
G.f.: A(x) = 1/(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012
G.f.: A(x) = 1 + x*d/dx { log( G(x)^5/(1+x*G(x)^2) )/2 }, where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 04 2012
G.f.: A(x) = 1/(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Jun 16 2013
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],2). - Peter Luschny, May 19 2015
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n)). - Ilya Gutkovskiy, Oct 25 2017
From G. C. Greubel, Feb 22 2021: (Start)
a(n) = Sum_{k=0..n} A171822(n, k).
a(n) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1). (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Apr 05 2024
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
G.f.: g^2/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)
G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025

A183161 Self-convolution equals A183160.

Original entry on oeis.org

1, 1, 5, 26, 145, 841, 5006, 30350, 186537, 1158685, 7258145, 45779420, 290399030, 1851032314, 11847434810, 76100034106, 490343021881, 3168174174105, 20520045125681, 133197288251330, 866293102078525, 5644234561103785
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2010

Keywords

Comments

Conjecture: a(n) is never congruent to 3 modulo 4; see A218622. - Paul D. Hanna, Nov 03 2012

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 + 13820*x^6 + 86662*x^7 +...+ A183160(n)*x^n +...

Links

Crossrefs

Cf. A183160, A218622, A001764, A002426, A219197.

Programs

  • PARI
    a(n)=local(A2=sum(m=0,n,sum(k=0,m,binomial(m+k,m-k)*binomial(2*m-k,k))*x^m+x*O(x^n)));polcoeff(A2^(1/2),n)
  • PARI
    a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/sqrt(1-2*x*G^2-3*x^2*G^4), n) \\ Paul D. Hanna, Nov 03 2012
  • PARI
    A002426(n)=sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))
{a(n)=if(n==0,1,sum(k=0,n,A002426(k)*binomial(3*n-k,n-k)*2*k/(3*n-k)))} \\ Paul D. Hanna, Nov 04 2012
for(n=0,30,print1(a(n),", "))
  • PARI
    a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/sqrt(1+3*x*G-5*x*G^2), n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2013

Formula

Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k) = A183160(n).
G.f.: A(x) = 1/sqrt(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012
G.f.: A(x) = Sum_{n>=0} A002426(n) * x^n * G(x)^(2*n), where A002426 are the central trinomial coefficients and G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012
a(n) = Sum_{k=0..n} A002426(k) * C(3*n-k,n-k) * 2*k/(3*n-k) for n>0, where A002426 are the central trinomial coefficients: A002426(n) = Sum_{k=0..[n/2]} C(n,2*k)*C(2*k,k). - Paul D. Hanna, Nov 04 2012
G.f.: A(x) = 1/sqrt(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Jun 16 2013
From Vaclav Kotesovec, Oct 05 2020: (Start)
Recurrence: 32*(n-1)*n*(2*n - 1)*(294*n^2 - 1253*n + 1310)*a(n) = 4*(n-1)*(62328*n^4 - 391468*n^3 + 870242*n^2 - 806673*n + 264534)*a(n-1) - 6*(132300*n^5 - 1169784*n^4 + 4019115*n^3 - 6676447*n^2 + 5328996*n - 1620540)*a(n-2) - 81*(n-2)*(3*n - 8)*(3*n - 7)*(294*n^2 - 665*n + 351)*a(n-3).
a(n) ~ 3^(3*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(2*n + 3/2)). (End)
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