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 1-3 of 3 results.

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

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

A361127 Let p = n-th odd prime; a(n) = index where 2*p appears in A360519, or -1 if 2*p never appears.

Original entry on oeis.org

2, 3, 11, 16, 28, 24, 32, 40, 48, 51, 55, 59, 84, 96, 104, 120, 123, 127, 144, 148, 160, 164, 176, 200, 203, 207, 208, 211, 236, 252, 260, 276, 280, 304, 308, 312, 332, 336, 344, 368, 376, 388, 392, 400, 404, 428, 452, 468, 472, 480, 496, 500, 508, 520, 532, 556, 560
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 08 2023

Keywords

Comments

It is conjectured that every 2*prime(n) (n>1) appears in A360519. A proof of this would be a big step towards proving that every term of C appears in A360519.

Examples

			p = 11 is the 4th odd prime, and A360519(16) = 2*11 = 22, so a(4) = 16.

Links

Crossrefs

Cf. A360519, A360103.
Showing 1-3 of 3 results.

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