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.

Showing 1-4 of 4 results.

A078995 a(n) = Sum_{k=0..n} C(4*k,k)*C(4*(n-k),n-k).

Original entry on oeis.org

1, 8, 72, 664, 6184, 57888, 543544, 5113872, 48180456, 454396000, 4288773152, 40503496536, 382701222296, 3617396099936, 34203591636048, 323492394385824, 3060238763412072, 28955508198895584, 274018698082833760, 2593539713410178528, 24550565251665845664
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2003

Keywords

Links

Crossrefs

See A049235 for more information.
Cf. A006256, A079563, A079678, A079679.
Cf. A005810, A147855, A226733, A226761, A385605, A386811.

Programs

  • Maple
    series(eval(g/(3*g-4), g=RootOf(g = 1+x*g^4,g))^2, x=0, 30); # Mark van Hoeij, May 06 2013
  • Mathematica
    Table[Sum[Binomial[4*k, k]*Binomial[4*(n - k), n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2012 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*k, k)*binomial(4*(n-k), n-k)); \\ Michel Marcus, May 09 2020

Formula

a(n) = 2/3*(256/27)^n*(1+c/sqrt(n)+o(n^-1/2)) where c = 2/3*sqrt(2/(3*Pi)) = 0.307105910641187... More generally, a(n, m)=sum(k=0, n, binomial(m*k, k)*binomial(m*(n-k), n-k)) is asymptotic to 1/2*m/(m-1)*(m^m/(m-1)^(m-1))^n. See A000302, A006256 for cases m=2 and 3. - Benoit Cloitre, Jan 26 2003, corrected and extended by Vaclav Kotesovec, Nov 06 2012
243*n*(8*n - 17)*(3*n - 1)*(3*n - 4)*(3*n - 2)*(3*n - 5)*a(n) = 72*(3*n - 5)*(3*n - 4)*(6912*n^4 - 33120*n^3 + 58256*n^2 - 47798*n + 15309)*a(n - 1) - 3072*(2*n - 3)*(6912*n^5 - 55008*n^4 + 175696*n^3 - 282180*n^2 + 227825*n - 73710)*a(n - 2) + 262144*(n - 2)*(4*n - 7)*(2*n - 3)*(2*n - 5)*(4*n - 9)*(8*n - 9)*a(n - 3). - Vladeta Jovovic, Jul 16 2004
Shorter recurrence: 81*n*(3*n-2)*(3*n-1)*(8*n-11)*a(n) = 24*(4608*n^4-14400*n^3+15776*n^2-7346*n+1215)*a(n-1) - 2048*(2*n-3)*(4*n-5)*(4*n-3)*(8*n-3)*a(n-2). - Vaclav Kotesovec, Nov 06 2012
a(n) = Sum_{k=0..n} binomial(4*k+l,k) * binomial(4*(n-k)-l,n-k) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 4^(n-k) * binomial(3*n+k,k). (End)
G.f.: g^2/(3*g-4)^2 where g=ogf(A002293) satisfies g = 1+x*g^4. - Mark van Hoeij, May 06 2013
a(n) = [x^n] 1/((1-4*x) * (1-x)^(3*n+1)). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 15 2025

A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

Original entry on oeis.org

1, 6, 46, 378, 3214, 27896, 245506, 2182396, 19548046, 176142312, 1594831736, 14497410186, 132224930146, 1209397179048, 11088872706188, 101890087382168, 937973964234638, 8649109175873288, 79872298511230120, 738583466508887304, 6837944227813170424
Offset: 0

Views

Author

Seiichi Manyama, Aug 03 2025

Keywords

Links

Crossrefs

Cf. A000302, A160906, A386812.
Cf. A005810, A078995, A147855, A226733, A226761, A385605.
Cf. A098430, A383716.
Cf. A066381.

Programs

  • Magma
    [&+[Binomial(4*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 21 2025
  • Mathematica
    Table[Sum[Binomial[4*n+1,k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n+1, k));

Formula

a(n) = [x^n] 1/((1-2*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k,k).
D-finite with recurrence +645*n*(3*n-1)*(3*n-2)*a(n) +8*(-56722*n^3+213090*n^2-305978*n+150255)*a(n-1) +128*(62908*n^3-282348*n^2+385070*n-126735)*a(n-2) +12288*(-2486*n^3+8918*n^2+758*n-18935)*a(n-3) -2949120*(2*n-7)*(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 03 2025
a(n) = 2^(4*n+1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 07 2025
a(n) ~ 2^(8*n + 3/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Aug 07 2025
G.f.: g^2/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/2), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(8-2*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 16 2025

A147855 G.f.: 1 / (1 + 4*x*G(x)^2 - 7*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 3, 22, 174, 1444, 12323, 107104, 942952, 8381596, 75053100, 676017962, 6118171326, 55591175956, 506805088026, 4633571685968, 42468065811884, 390071875757852, 3589637747968964, 33089300640166360, 305476314574338648, 2823932709938708824, 26137341654281261347
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 22*x^2 + 174*x^3 + 1444*x^4 + 12323*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 4*x*G(x)^2 - 7*x*G(x)^3).

Links

Crossrefs

Cf. A226705, A002293.
Cf. A026641, A183160, A371753.
Cf. A005810, A078995, A226733, A226761, A385605, A386811.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n+k,n-k]*Binomial[2*n-k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(2*n+k, n-k)*binomial(2*n-k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(k, n-k)*binomial(4*n-k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+4*x*G^2-7*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-3*x*G^2-7*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} C(k, n-k) * C(4*n-k, k).
a(n) = Sum_{k=0..n} C(n+k, n-k) * C(3*n-k, k).
a(n) = Sum_{k=0..n} C(2*n+k, n-k) * C(2*n-k, k).
a(n) = Sum_{k=0..n} C(3*n+k, n-k) * C(n-k, k).
a(n) = Sum_{k=0..n} C(4*n+k, n-k) * C(-k, k).
G.f.: 1 / (1 - 3*x*G(x)^2 - 7*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+5/2)/(5*sqrt(Pi*n)*3^(3*n+1/2)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Apr 05 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1-x^2) * (1-x)^(3*n)). (End)
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-1+2*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 5*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A226733 G.f.: 1 / (1 + 8*x*G(x)^2 - 10*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 18, 142, 1186, 10152, 88414, 779508, 6936066, 62159224, 560238728, 5072970366, 46114086446, 420558296888, 3846232573236, 35261290343112, 323952686556354, 2981787128165592, 27491128592627800, 253835886034173848, 2346892194318851016, 21724880414632781472
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 142*x^3 + 1186*x^4 + 10152*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 8*x*G(x)^2 - 10*x*G(x)^3).

Links

Crossrefs

Cf. A147855, A183160, A226705, A002293.
Cf. A005810, A078995, A147855, A226761, A385605, A386811.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n+2*k,n-k]*Binomial[2*n-2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+8*x*G^2-10*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-2*x*G^2-10*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(2*n+2*k, n-k)*binomial(2*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(4*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(4*n+2*k, n-k)*binomial(-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} C(2*k, n-k) * C(4*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(-2*k, k).
G.f.: 1 / (1 - 2*x*G(x)^2 - 10*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+3/2)/(3^(3*n+3/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-2+3*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 3*(B(x)-1)/2), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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