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.

A162477 Expansion of (1/(1-x)^2)*c(x/(1-x)^4) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 3, 11, 50, 255, 1391, 7939, 46821, 283081, 1745212, 10929625, 69338213, 444668749, 2877994064, 18774736487, 123321704739, 814930698217, 5413955476648, 36138368789601, 242252716083298, 1630170332414433
Offset: 0

Views

Author

Paul Barry, Jul 04 2009

Keywords

Comments

Partial sums of A162476.

Links

Crossrefs

Cf. A000108, A086616, A162476.

Programs

  • Mathematica
    Table[Sum[Binomial[n+3k+1,4k+1]CatalanNumber[k],{k,0,n}],{n,0,100}] (* Emanuele Munarini, Aug 31 2017 *)
  • Maxima
    makelist(sum(binomial(n+3*k+1,4*k+1)*binomial(2*k,k)/(k+1),k,0,n),n,0,12); /* Emanuele Munarini, Aug 31 2017 */

Formula

G.f.: 1/((1 - x)^2 - x/((1 - x)^2 - x/((1 - x)^2 - x/((1 - x)^2 - ... (continued fraction);
a(n) = Sum_{k=0..n} C(n+3k+1,n-k)*A000108(k).
Conjecture: (n+1)*a(n) +4*(1-2*n)*a(n-1) +6*(n-2)*a(n-2) +2*(7-2*n)*a(n-3) +(n-5)*a(n-4) = 0. - R. J. Mathar, Nov 17 2011
G.f.: (1 - 2*x + x^2 - sqrt(1 - 8*x + 6*x^2 - 4*x^3 + x^4))/(2*x). Remark: using this form of the g.f., it is straightforward to prove the above conjectured recurrence. - Emanuele Munarini, Aug 31 2017
G.f. A(x) satisfies: A(x) = (1 + x*A(x)^2) / (1 - x)^2. - Ilya Gutkovskiy, Jun 30 2020
G.f.: 1/G(x), where G(x) = 1 - (3*x - x^2)/(1 - x/G(x)) (continued fraction). - Nikolaos Pantelidis, Jan 08 2023

A162475 Expansion of c(x/(1-x)^4), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 6, 31, 166, 931, 5412, 32334, 197378, 1225871, 7722282, 49224175, 316921948, 2057994779, 13463417108, 88650225829, 587062025226, 3907415784953, 26125388534522, 175389933980744, 1181803269037438, 7989829660805193
Offset: 0

Views

Author

Paul Barry, Jul 04 2009

Keywords

Crossrefs

Cf. A000108.
Partial sums are A162476.

Formula

G.f.: 1/(1-x/((1-x)^4-x/(1-x/((1-x)^4-x/(1-... (continued fraction);
a(n) = Sum_{k=0..n} C(n+3k-1,n-k)*A000108(k).
Conjecture: (n+1)*a(n) +3*(2-3n)*a(n-1) +2*(7n-20)*a(n-2) +2*(22-5n)*a(n-3) +(5n-31)*a(n-4) +(8-n)*a(n-5)=0. - R. J. Mathar, Nov 17 2011

A360102 a(n) = Sum_{k=0..n} binomial(n+2*k,n-k) * Catalan(k).

Original entry on oeis.org

1, 2, 7, 30, 141, 703, 3655, 19603, 107679, 602756, 3426049, 19721069, 114728723, 673494466, 3984493735, 23732956453, 142204128507, 856560123504, 5183708936061, 31502904805922, 192180259402691, 1176416604202925, 7223943302003917, 44486888142708088
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums of A360100.
Partial sums are A258973.
Cf. A000108, A162481.
Cf. A006318, A007317, A162476, A360103.

Programs

  • Maple
    A360102 := proc(n)
    add(binomial(n+2*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360102(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n, binomial(n+2*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)^3))))

Formula

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^2.
G.f.: (1/(1-x)) * c(x/(1-x)^3), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +10*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +(n-5)*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
Showing 1-4 of 4 results.

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