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.

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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

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).

A358050 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j) * binomial(k*(n-j),n-j).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 16, 4, 0, 1, 8, 39, 64, 5, 0, 1, 10, 72, 258, 256, 6, 0, 1, 12, 115, 664, 1719, 1024, 7, 0, 1, 14, 168, 1360, 6184, 11496, 4096, 8, 0, 1, 16, 231, 2424, 16265, 57888, 77052, 16384, 9, 0, 1, 18, 304, 3934, 35400, 195660, 543544, 517194, 65536, 10, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2022

Keywords

Examples

			Square array begins:
  1, 1,    1,     1,     1,      1, ...
  0, 2,    4,     6,     8,     10, ...
  0, 3,   16,    39,    72,    115, ...
  0, 4,   64,   258,   664,   1360, ...
  0, 5,  256,  1719,  6184,  16265, ...
  0, 6, 1024, 11496, 57888, 195660, ...

Crossrefs

Column k=0-7 give: A000007, A001477(n+1), A000302, A006256, A078995, A079678, A079679, A079563.
Main diagonal gives A358145.
Cf. A358146.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(k*j, j)*binomial(k*(n-j), n-j));
  • PARI
    T(n, k) = sum(j=0, n, (k-1)^(n-j)*binomial(k*n+1, j));
  • PARI
    T(n, k) = sum(j=0, n, k^(n-j)*binomial((k-1)*n+j, j));

Formula

T(n,k) = Sum_{j=0..n} (k-1)^(n-j) * binomial(k*n+1,j).
T(n,k) = Sum_{j=0..n} k^(n-j) * binomial((k-1)*n+j,j).

A386371 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n+1,k).

Original entry on oeis.org

1, 3, 31, 317, 3399, 37418, 419229, 4756104, 54463335, 628197809, 7287712566, 84942987198, 993941174829, 11668806723876, 137378189197112, 1621322803014672, 19175540677541991, 227217662222902443, 2696878158795639549, 32057403690640189635, 381573145993865438254
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2025

Keywords

Links

Crossrefs

Cf. A001449, A079589, A079678, A371753, A385632, A386812.
Cf. A226705, A226733, A226751, A387085.
Cf. A002294.

Programs

  • Magma
    [&+[(-3)^(n-k) * Binomial(5*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[(-3)^(n-k)*Binomial[5*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n+1, k));

Formula

a(n) = [x^n] (1+x)^(5*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(4*n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(5*n-k,n-k).
G.f.: 1/(1 - x*g^3*(-10+13*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g^2/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 7*(B(x)-1)/5), where B(x) is the g.f. of A001449.
D-finite with recurrence 648*n*(135551509682187347695*n -244103380745409504343) *(4*n-1)*(2*n-1)*(4*n-3)*a(n) +(-33979500619583537984836075*n^5 +130803893690808003041848009*n^4 -168380151442376797602371231*n^3 +62069291513227826684567999*n^2 +49760069127090078338544954*n -39530305857276050670355320)*a(n-1) +40*(-108999332467309598098777*n^5 -28981701912184019189355*n^4 -1554974299825191814369159*n^3 +13581461461293413639358363*n^2 -28599284433109723900055776*n +18909354537435947334628944)*a(n-2) +211200*(5*n-11) *(5*n-9)*(28440609019752807*n +93502568692163852)*(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 26 2025
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