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.

A386614 a(n) = Sum_{k=0..n-1} binomial(5*k+1,k) * binomial(5*n-5*k,n-k-1).

Original entry on oeis.org

0, 1, 16, 220, 2880, 36850, 465536, 5834852, 72744640, 903525715, 11191199200, 138323478980, 1706860996096, 21034268215120, 258934785258240, 3184696786012500, 39140208951032960, 480734044749851305, 5901368553964031600, 72410017973538837880, 888114187330722044800, 10888921795007470528060
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2025

Keywords

Crossrefs

Cf. A079678, A386367, A386566, A386613.
Cf. A006419, A386612, A386616, A386617.

Programs

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

Formula

G.f.: g^3 * (g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/((1-g)^2 * (1-5*g)^2) where g*(1-g)^4 = x.
a(n) = Sum_{k=0..n-1} binomial(5*k+1+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+2,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k+2,k).
D-finite with recurrence +35651584*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) +8192*(56348704*n^4-268019168*n^3+418502324*n^2-264019618*n+57303885)*a(n-1) +160*(-65524820000*n^4+314102050000*n^3-463341186250*n^2+159732814775*n+76118151939)*a(n-2) +62500*(660806875*n^4-1813661250*n^3-5080986250*n^2+20705993100*n-17279228304)*a(n-3) +308935546875*(5*n-11)*(5*n-14)*(5*n-13)*(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 10 2025

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).

A386566 a(n) = Sum_{k=0..n-1} binomial(5*k-1,k) * binomial(5*n-5*k,n-k-1).

Original entry on oeis.org

0, 1, 14, 181, 2284, 28506, 353630, 4370584, 53882392, 663116347, 8150224204, 100073884670, 1227826127020, 15055154471696, 184508186225552, 2260299193652496, 27679951219660080, 338872887728053465, 4147618793911034330, 50753529798492061819, 620942367878256638264
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2025

Keywords

Examples

			(1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ) = x + 7*x^2 + 181*x^3/3 + 571*x^4 + 28506*x^5/5 + ...

Crossrefs

Cf. A000346, A062236, A386565, A386567.
Cf. A079678, A386367, A386613, A386614.
Cf. A002294, A118971.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(5*k-1, k)*binomial(5*n-5*k, n-k-1));
  • PARI
    my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(5*k, k)/(4*k+1)*x^k)); concat(0, Vec(g*(g-1)/(5-4*g)^2))

Formula

G.f.: g*(g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(5*k-1+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,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k,k).
Conjecture D-finite with recurrence 196608*n*(4*n-3)*(2*n-1)*(18270873280*n -32560150837) *(4*n-1)*a(n) +1280*(-1399185802400000*n^5 +1022280893000000*n^4 +17669158913120000*n^3 -48968110172924750*n^2 +49502057719349955*n -17877514345852392)*a(n-1) +125000*(-61298198200000*n^5 +1447969779032500*n^4 -7721498995066250*n^3 +17474948768595875*n^2 -18352567310653770*n +7399184154389181)*a(n-2) +48828125*(5*n-11) *(5*n-14)*(4958243695*n -6717884799) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

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).

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.

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