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.

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).

A386368 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k-2,n-k-1).

Original entry on oeis.org

0, 1, 16, 246, 3736, 56421, 849432, 12763878, 191548464, 2871970110, 43031833656, 644432826478, 9646983339456, 144366433138955, 2159869510669320, 32306874783230556, 483151884326658144, 7224464127509984490, 108011596038055519680, 1614676987907480393940
Offset: 0

Views

Author

Seiichi Manyama, Jul 19 2025

Keywords

Examples

			(1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ) = x + 8*x^2 + 82*x^3 + 934*x^4 + 56421*x^5/5 + ...

Crossrefs

Cf. A000302, A036829, A308523, A386367.
Cf. A079679, A386567, A386615, A386616.
Cf. A002295, A130564.

Programs

  • Maple
    A386368 := proc(n::integer)
    add(binomial(6*k,k)*binomial(6*n-6*k-2,n-k-1),k=0..n-1) ;
end proc:
seq(A386368(n),n=0..80) ; # R. J. Mathar, Jul 30 2025
  • PARI
    a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1));
  • PARI
    my(N=20, x='x+O('x^N), g=x*sum(k=0, N, binomial(6*k+4, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-6*g)^2))

Formula

G.f.: g*(1-g)/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ).
G.f.: (g-1)/(6-5*g)^2 where g=1+x*g^6.
a(n) = Sum_{k=0..n-1} binomial(6*k-2+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).
Conjecture D-finite with recurrence 48828125*(n-1)*(5*n-4)*(5*n-3) *(432862082629612805*n -769306661967834399) *(5*n-2)*(5*n-1)*a(n) +1125000*(-405245406115816219575000*n^6 +2613180799468910510392500*n^5 -7667164406968651479521250*n^4 +13834502135358262506660375*n^3 -16251583347734702117341345*n^2 +11251247074043948959380314*n -3395699069351241765495720)*a(n-1) +33592320*(142281690918326440537500*n^6 -1266424338521609272012500*n^5 +5236041263583271687953750*n^4 -12786608152035075786775875*n^3 +18838556229131595646260055*n^2 -15323925851720394901667853*n +5240681406952416812161236)*a(n-2) -53444359913472*(6*n-17) *(395547729523405*n -538181211711288)*(6*n-13) *(3*n-7)*(2*n-5) *(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

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).

A386611 a(n) = Sum_{k=0..n-1} binomial(4*k,k) * binomial(4*n-4*k,n-k-1).

Original entry on oeis.org

0, 1, 12, 126, 1268, 12513, 122148, 1184364, 11432100, 109997460, 1055891248, 10117633542, 96812495820, 925334377822, 8836315646616, 84317468847768, 804064275489924, 7663595943744876, 73009005101019792, 695263276434909976, 6618709687608909648, 62989317586872238689
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2025

Keywords

Crossrefs

Cf. A308523, A386565, A386612.
Cf. A008549, A386613, A386615.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(4*k, k)*binomial(4*n-4*k, n-k-1));

Formula

G.f.: g^2 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g) * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+1,k).

A386613 a(n) = Sum_{k=0..n-1} binomial(5*k,k) * binomial(5*n-5*k,n-k-1).

Original entry on oeis.org

0, 1, 15, 200, 2570, 32470, 406411, 5057440, 62692100, 775007135, 9561421830, 117780193480, 1449107627450, 17811990468400, 218768774024360, 2685209277718320, 32940971570389960, 403920568087927025, 4950915045235523125, 60663591616305306320, 743092566613017730980, 9100088494955802407060
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2025

Keywords

Crossrefs

Cf. A079678, A386367, A386566, A386614.
Cf. A008549, A386611, A386615.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(5*k, k)*binomial(5*n-5*k, n-k-1));

Formula

G.f.: g^2 * (g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/((1-g) * (1-5*g)^2) where g*(1-g)^4 = x.
a(n) = Sum_{k=0..n-1} binomial(5*k+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).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.