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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A360100 a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 5, 23, 111, 562, 2952, 15948, 88076, 495077, 2823293, 16295020, 95007654, 558765743, 3310999269, 19748462718, 118471172054, 714355994997, 4327148812557, 26319195869861, 160677354596769, 984236344800234, 6047526697800992, 37262944840704171
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums are A360102.
Cf. A162481, A258973.
Cf. A002212, A006319, A162475, A360101.
Cf. A000108.

Programs

  • Maple
    A360100 := proc(n)
    add(binomial(n+2*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360100(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    m = 24;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^3 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+2*k-1, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^3)))

Formula

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^3.
G.f.: c(x/(1-x)^3), where c(x) is the g.f. of A000108.
a(n) ~ sqrt(-2 + (35 - 3*sqrt(129))^(1/3) + (35 + 3*sqrt(129))^(1/3)) * (((7 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / (sqrt(2*Pi) * n^(3/2))). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence (n+1)*a(n) +(-8*n+5)*a(n-1) +(10*n-27)*a(n-2) +(-4*n+17)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A360103 a(n) = Sum_{k=0..n} binomial(n+4*k,n-k) * Catalan(k).

Original entry on oeis.org

1, 2, 9, 49, 283, 1715, 10793, 69906, 463031, 3122264, 21363065, 147951489, 1035173405, 7306326465, 51959150713, 371950057003, 2678083379707, 19381867703946, 140915907625531, 1028760981192771, 7538511404971231, 55427326349613665, 408789584900354397
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums of A360101.
Cf. A000108, A360057.
Cf. A006318, A007317, A162476, A360102.

Programs

  • Maple
    A360103 := proc(n)
    add(binomial(n+4*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360103(n),n=0..40) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^5))))

Formula

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^4.
G.f.: (1/(1-x)) * c(x/(1-x)^5), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +2*(-5*n+3)*a(n-1) +(19*n-47)*a(n-2) +20*(-n+4)*a(n-3) +5*(3*n-17)*a(n-4) +2*(-3*n+22)*a(n-5) +(n-9)*a(n-6)=0. - R. J. Mathar, Mar 12 2023
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