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

A386565 a(n) = Sum_{k=0..n-1} binomial(4*k-1,k) * binomial(4*n-4*k,n-k-1).

Original entry on oeis.org

0, 1, 11, 111, 1091, 10596, 102237, 982458, 9415539, 90063180, 860278156, 8208539351, 78258171957, 745595635084, 7099714918062, 67574576298276, 642927956583123, 6115089154367484, 58146652079312580, 552769690436583532, 5253812277363417836, 49925987913040522128
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2025

Keywords

Examples

			(1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ) = x + 11*x^2/2 + 37*x^3 + 1091*x^4/4 + 10596*x^5/5 + ...

Crossrefs

Cf. A000346, A062236, A386566, A386567.
Cf. A002293, A006632, A308523.

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(4*k-1, k)*binomial(4*n-4*k, n-k-1));
  • PARI
    my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); concat(0, Vec(g*(g-1)/(4-3*g)^2))

Formula

G.f.: g*(g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(4*k-1+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,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k,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).

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.

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