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.

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

A360101 a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 7, 40, 234, 1432, 9078, 59113, 393125, 2659233, 18240801, 126588424, 887221916, 6271153060, 44652824248, 319990906290, 2306133322704, 16703784324239, 121534039921585, 887845073567240, 6509750423778460, 47888814944642434, 353362258550740732
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums are A360103.
Cf. A000108, A360057.
Cf. A002212, A006319, A162475, A360100.

Programs

  • Maple
    A360101 := proc(n)
    add(binomial(n+4*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360101(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    m = 23;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^5 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+4*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)^5)))

Formula

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^5.
G.f.: c(x/(1-x)^5), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +(-10*n+7)*a(n-1) +(19*n-56)*a(n-2) +10*(-2*n+9)*a(n-3) +5*(3*n-19)*a(n-4) +(-6*n+49)*a(n-5) +(n-10)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

A358518 a(n) = Sum_{k=0..n} binomial(n+3*k+3,n-k) * Catalan(k).

Original entry on oeis.org

1, 5, 20, 85, 405, 2116, 11766, 68237, 407789, 2492553, 15506942, 97859544, 624880895, 4029896310, 26209648212, 171711104853, 1132143259711, 7506530891217, 50019287312324, 334784759816729, 2249720564735567, 15172573979205166, 102662981205576494
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A086616, A162481, A360057.
Cf. A000108, A360046.

Programs

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

Formula

a(n) = binomial(n+3,3) + Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^4 + x * A(x)^2.
G.f.: 2 / ( (1-x)^4 * (1 + sqrt( 1 - 4*x/(1-x)^4 )) ).
D-finite with recurrence (n+1)*a(n) +(-9*n+2)*a(n-1) +2*(7*n-4)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

A360060 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+4*k+4,n-k) * Catalan(k).

Original entry on oeis.org

1, 4, 7, 5, 4, 29, 50, -83, -185, 743, 1425, -5250, -9868, 40530, 73280, -319155, -557485, 2573032, 4341065, -21107670, -34398290, 175655925, 276438452, -1479202280, -2247154681, 12581036223, 18440253397, -107916225837, -152514334540, 932452267956, 1269723550920
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A360058, A360059.
Cf. A000108, A360051, A360057.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(2/((1-x)^5*(1+sqrt(1+4*x/(1-x)^5))))

Formula

a(n) = binomial(n+4,4) - Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^5 - x * A(x)^2.
G.f.: 2 / ( (1-x)^5 * (1 + sqrt( 1 + 4*x/(1-x)^5 )) ).
D-finite with recurrence (n+1)*a(n) +2*(-n-1)*a(n-1) +(11*n-19)*a(n-2) +20*(-n+2)*a(n-3) +15*(n-3)*a(n-4) +6*(-n+4)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023
Showing 1-4 of 4 results.

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