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.

A360049 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+2,3*k+2) * Catalan(k).

Original entry on oeis.org

1, 3, 6, 9, 9, 0, -26, -72, -117, -82, 204, 975, 2289, 3357, 1332, -9834, -37935, -82593, -108282, 2583, 487521, 1621071, 3261546, 3685230, -2318615, -24607854, -72887472, -134909701, -123941901, 200330184, 1258932996, 3377359872, 5706502677, 3797618237
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Links

Crossrefs

Cf. A360048, A360050, A360051.
Cf. A000108.

Programs

  • Mathematica
    Table[Sum[(-1)^k Binomial[n+2,3k+2]CatalanNumber[k],{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Nov 21 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(n+2, 3*k+2)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(2/((1-x)*((1-x)^2+sqrt((1-x)^4+4*x^3*(1-x)))))

Formula

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

A360050 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n+3,4*k+3) * Catalan(k).

Original entry on oeis.org

1, 4, 10, 20, 34, 48, 48, 0, -163, -548, -1274, -2340, -3255, -2224, 5304, 28560, 82379, 182300, 322102, 410700, 133128, -1295264, -5440600, -14733680, -31384533, -52870668, -59633454, 11449780, 312532426, 1137823168, 2918752832, 5961965824, 9464314955
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A360048, A360049, A360051.
Cf. A000108.

Programs

  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n+3, 4*k+3)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(2/((1-x)^2*((1-x)^2+sqrt((1-x)^4+4*x^4))))

Formula

a(n) = binomial(n+3,3) - Sum_{k=0..n-4} a(k) * a(n-k-4).
G.f. A(x) satisfies: A(x) = 1/(1-x)^4 - x^4 * A(x)^2.
G.f.: 2 / ( (1-x)^2 * ((1-x)^2 + sqrt((1-x)^4 + 4*x^4)) ).
D-finite with recurrence (n+4)*a(n) +5*(-n-3)*a(n-1) +10*(n+2)*a(n-2) +10*(-n-1)*a(n-3) +(9*n-8)*a(n-4) +5*(-n+1)*a(n-5) =0. - R. J. Mathar, Jan 25 2023

A360048 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+1,2*k+1) * Catalan(k).

Original entry on oeis.org

1, 2, 2, 0, -3, -2, 9, 24, 11, -66, -152, -8, 587, 1082, -438, -5248, -7733, 7942, 47502, 53792, -105313, -430118, -343043, 1249800, 3866557, 1730018, -13996096, -34243896, -1947203, 150962374, 296101865, -121857184, -1582561869, -2468098042, 2529520766
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A360049, A360050, A360051.
Cf. A000108.

Programs

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

Formula

a(n) = n+1 - Sum_{k=0..n-2} a(k) * a(n-k-2).
G.f. A(x) satisfies: A(x) = 1/(1-x)^2 - x^2 * A(x)^2.
G.f.: 2 / ( (1-x) * (1-x + sqrt((1-x)^2 + 4*x^2)) ).
D-finite with recurrence (n+2)*a(n) +3*(-n-1)*a(n-1) +(7*n-4)*a(n-2) +5*(-n+1)*a(n-3)=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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