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 1-5 of 5 results.

A036829 a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).

Original entry on oeis.org

0, 1, 7, 48, 327, 2221, 15060, 102012, 690519, 4671819, 31596447, 213633696, 1444131108, 9760401756, 65957919496, 445671648228, 3011064814455, 20341769686311, 137412453018933, 928188965638464, 6269358748632207, 42343731580741821
Offset: 0

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Author

N. J. A. Sloane

Keywords

References

  • M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Links

Crossrefs

Cf. A006256.
Cf. A001764, A006013, A007318, A036917, A062236.
Cf. A000302, A308523, A386367, A386368.

Programs

  • Haskell
    a036829 n = sum $ map
   (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
-- Reinhard Zumkeller, May 24 2012
  • Mathematica
    Table[Sum[Binomial[3k,k]Binomial[3n-3k-2,n-k-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Jan 10 2012 *)

Formula

G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
Recurrence: 8*(n-1)*(2*n-1)*a(n) = 6*(36*n^2-81*n+49)*a(n-1) - 81*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 3^(3*n-1)/2^(2*n+1). - Vaclav Kotesovec, Dec 29 2012
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(3*k,k)*x^k ). - Seiichi Manyama, Jul 19 2025
G.f.: (g-1)/(3-2*g)^2 where g=1+x*g^3. - Seiichi Manyama, Jul 26 2025

A386616 a(n) = Sum_{k=0..n-1} binomial(6*k+1,k) * binomial(6*n-6*k,n-k-1).

Original entry on oeis.org

0, 1, 19, 315, 5000, 77785, 1196667, 18282742, 278031900, 4214278350, 63723788295, 961789682008, 14495501585664, 218216042892175, 3281961694927950, 49322417450239980, 740753733463215604, 11118981305235476010, 166821561372208253850, 2501861335268901337425, 37507747177968865536840
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2025

Keywords

Crossrefs

Cf. A079679, A386368, A386567, A386615.
Cf. A006419, A386612, A386614, A386617.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(6*k+1, k)*binomial(6*n-6*k, n-k-1));

Formula

G.f.: g^3 * (g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/((1-g)^2 * (1-6*g)^2) where g*(1-g)^5 = x.
a(n) = Sum_{k=0..n-1} binomial(6*k+1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n+2,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k+2,k).

A386367 a(n) = Sum_{k=0..n-1} binomial(5*k,k) * binomial(5*n-5*k-2,n-k-1).

Original entry on oeis.org

0, 1, 13, 163, 2021, 24930, 306655, 3765448, 46182101, 565939603, 6931070490, 84845250370, 1038235255415, 12700966517968, 155336699256808, 1899439862390640, 23222289820948405, 283872591297526505, 3469680960837171415, 42404345427419774621, 518193229118757697930
Offset: 0

Views

Author

Seiichi Manyama, Jul 19 2025

Keywords

Examples

			(1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ) = x + 13*x^2/2 + 163*x^3/3 + 2021*x^4/4 + 4986*x^5 + ...

Crossrefs

Cf. A000302, A036829, A308523, A386368.
Cf. A079678, A386566, A386613, A386614.
Cf. A002294, A118971.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(5*k, k)*binomial(5*n-5*k-2, n-k-1));
  • PARI
    my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(5*k+3, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-5*g)^2))

Formula

G.f.: g*(1-g)/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ).
G.f.: (g-1)/(5-4*g)^2 where g=1+x*g^5.
a(n) = Sum_{k=0..n-1} binomial(5*k-2+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n-1,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k-1,k).

A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).

Original entry on oeis.org

0, 1, 17, 268, 4129, 62955, 954392, 14417376, 217279857, 3269099590, 49125066135, 737516631908, 11064270530632, 165889863957065, 2486052264852180, 37241727274394640, 557707191712371729, 8349517132932620730, 124971965902300790390, 1870139909398530770760
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2025

Keywords

Examples

			(1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ) = x + 17*x^2/2 + 268*x^3/3 + 4129*x^4/4 + 12591*x^5 + ...

Crossrefs

Cf. A000346, A062236, A386565, A386566.
Cf. A079679, A386368, A386615, A386616.
Cf. A002295, A130564.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1));
  • PARI
    my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(6*k, k)/(5*k+1)*x^k)); concat(0, Vec(g*(g-1)/(6-5*g)^2))

Formula

G.f.: g*(g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(6*k-1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k,k).

A386615 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k,n-k-1).

Original entry on oeis.org

0, 1, 18, 291, 4550, 70065, 1069872, 16251694, 246010014, 3714826350, 55993450830, 842823848448, 12672667549488, 190381643518855, 2858101359683400, 42882348756992220, 643085584745669134, 9640075656634321770, 144457232389535563980, 2164044325920832653825, 32409930873969839549610
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2025

Keywords

Crossrefs

Cf. A079679, A386368, A386567, A386616.
Cf. A008549, A386611, A386613.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k, n-k-1));

Formula

G.f.: g^2 * (g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/((1-g) * (1-6*g)^2) where g*(1-g)^5 = x.
a(n) = Sum_{k=0..n-1} binomial(6*k+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n+1,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k+1,k).
Showing 1-5 of 5 results.

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

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