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 1-4 of 4 results.

A364371 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^2).

Original entry on oeis.org

1, 0, -1, 2, -2, -1, 9, -20, 20, 24, -150, 327, -293, -599, 3097, -6452, 4854, 15878, -71252, 140112, -81328, -437346, 1746254, -3214989, 1223971, 12345295, -44552833, 76242173, -11292089, -354175849, 1167638037, -1842585992, -233903034, 10273377388, -31169512310
Offset: 0

Views

Author

Seiichi Manyama, Jul 20 2023

Keywords

Crossrefs

Cf. A364372, A364373, A364375.
Cf. A000108, A073157.

Programs

  • Maple
    A364371 := proc(n)
    add((-1)^k* binomial(2*k+1,k) * binomial(2*k+1,n-k)/(2*k+1),k=0..n) ;
end proc:
seq(A364371(n),n=0..70); # R. J. Mathar, Jul 25 2023
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(2*k+1, k)*binomial(2*k+1, n-k)/(2*k+1));

Formula

G.f.: A(x) = 2*(1 + x) / (1 + sqrt(1+4*x*(1 + x)^2)).
a(n) = Sum_{k=0..n} (-1)^k * binomial(2*k+1,k) * binomial(2*k+1,n-k) / (2*k+1).
D-finite with recurrence (n+1)*a(n) +(5*n-1)*a(n-1) +6*(2*n-3)*a(n-2) +6*(2*n-5)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 24 2024: (Start)
A(x) = (1 + x)*c(-x*(1+x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
(1/x) * series_reversion(x/A(x)) = 1 - x^2 + 2*x^3 - 11*x^5 + 28*x^6 + ..., the g.f. of A364375. (End)

A364375 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^3).

Original entry on oeis.org

1, 0, -1, 2, 0, -11, 28, 1, -206, 564, 38, -4711, 13329, 1273, -119762, 344707, 41884, -3251250, 9445976, 1381154, -92305098, 269504686, 45848871, -2707126108, 7921304973, 1532928960, -81375728566, 238196143730, 51591751698, -2493907008116, 7293147604136
Offset: 0

Views

Author

Seiichi Manyama, Jul 21 2023

Keywords

Crossrefs

Cf. A364371, A364372, A364374, A364376.
Cf. A198953.

Programs

  • Maple
    A364375 := proc(n)
    add( (-1)^k*binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k)/(n+2*k+1),k=0..n) ;
end proc:
seq(A364375(n),n=0..80); # R. J. Mathar, Jul 25 2023
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+1, k)*binomial(n+2*k+1, n-k)/(n+2*k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1).
D-finite with recurrence +2*n*(191553133*n -462036810)*(2*n+1) *(n+1)*a(n) +2*n*(6735679202*n^3 -31340869996*n^2 +39568451245*n -13340358389)*a(n-1) +6*(13937077342*n^4 -106287464449*n^3 +278022830194*n^2 -296712736455*n +108876423952)*a(n-2) +6*(42118990776*n^4 -422141236704*n^3 +1546534911485*n^2 -2448212978721*n +1409411956166)*a(n-3) +6*(72631772298*n^4 -948761263665*n^3 +4512788370945*n^2 -9254886913710*n +6888712179986)*a(n-4) +3*(10147840245*n^4 -513806508936*n^3 +5519825354705*n^2 -22028093493130*n +30003008863784)*a(n-5) +6*(-9503341830*n^4 +235269814455*n^3 -2064338754902*n^2 +7709425316943*n -10409244067330)*a(n-6) +18*(3*n-20)*(n-6) *(156488131*n-746235854) *(3*n-13)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 24 2024: (Start)
G.f. A(x) satisfies (1/x) * series_reversion(x*A(x)) = 1/G(x), where G(x) is the g.f. of A364371.
P-recursive: fifth-order recurrence: (2*n+1)*(2*n+2)*(3045*n^5-26680*n^4+84901*n^3-123566*n^2+86300*n-25368)*n*a(n) + 6*(18270*n^7-160080*n^6+500851*n^5-666969*n^4+307749*n^3+70849*n^2-76222*n+8288)*n*a(n-1) + 6*(54810*n^8-562455*n^7+2191158*n^6-3956204*n^5+2960986*n^4+88959*n^3-1045774*n^2+187688*n+69888)*a(n-2) + 6*(109620*n^8-1289340*n^7+5897421*n^6-13016841*n^5+13725877*n^4-5967199*n^3+2484230*n^2-3359528*n+1002624)*a(n-3) - 6*(54810*n^8-726885*n^7+3719313*n^6-9080919*n^5+10367473*n^4-4378276*n^3+1152956*n^2-2297912*n+768096)*a(n-4) + (3*n-7)*(3*n-12)*(3*n-14)*(3045*n^5-11455*n^4+8631*n^3+1507*n^2+2376*n-1368)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 2 and a(4) = 0. (End)

A364376 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^4).

Original entry on oeis.org

1, 0, -1, 3, -4, -9, 73, -212, 111, 1956, -10078, 21466, 29823, -418183, 1561911, -1722963, -13205004, 86962328, -232448945, -109578204, 3849218852, -17135183489, 27800381006, 113891855632, -966644138742, 3075070731677, -833503324311, -41673632701038
Offset: 0

Views

Author

Seiichi Manyama, Jul 21 2023

Keywords

Crossrefs

Cf. A364372, A364374, A364375.
Cf. A215623.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+3*k+1, k)*binomial(n+3*k+1, n-k)/(n+3*k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+3*k+1,k) * binomial(n+3*k+1,n-k) / (n+3*k+1).
G.f.: x/series_reversion(x*G(x)), where G(x) = 1 - x^2 + 3*x^3 - 6*x^4 + 6*x^5 + 15*x^6 - ... is the g.f. of A364372. - Peter Bala, Aug 27 2024

A364373 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^4).

Original entry on oeis.org

1, 0, -1, 4, -12, 26, -14, -236, 1604, -6577, 17827, -14064, -186496, 1437856, -6416576, 18733256, -17358808, -201270728, 1652571996, -7692333934, 23375782030, -23913813710, -250917362258, 2147925544190, -10270145045142, 32053993413694, -35259817590134
Offset: 0

Views

Author

Seiichi Manyama, Jul 20 2023

Keywords

Crossrefs

Cf. A364371, A364372.
Cf. A364337.

Programs

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

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(4*k+1,k) * binomial(4*k+1,n-k) / (4*k+1).
Showing 1-4 of 4 results.

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