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.

Previous Showing 11-16 of 16 results.

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

A364621 G.f. satisfies A(x) = 1/(1-x)^2 + x*A(x)^4.

Original entry on oeis.org

1, 3, 15, 118, 1125, 11805, 131431, 1524090, 18208749, 222570985, 2770129627, 34985756752, 447243818573, 5775955923428, 75245253495035, 987627627396792, 13048147674230169, 173382031819242855, 2315662483861709467, 31068798980975635130, 418552735866147739185
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Crossrefs

Cf. A086616, A364620.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k+1, 6*k+1)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k+1,6*k+1) * binomial(4*k,k) / (3*k+1).

A336858 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1, k) + T(n-1,k-1) with T(n,0) = T(n, n) = 1 (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 9, 1, 1, 7, 21, 31, 1, 1, 9, 37, 89, 121, 1, 1, 11, 57, 183, 393, 515, 1, 1, 13, 81, 321, 897, 1805, 2321, 1, 1, 15, 109, 511, 1729, 4431, 8557, 10879, 1, 1, 17, 141, 761, 3001, 9161, 22149, 41585, 52465, 1, 1, 19, 177, 1079, 4841, 17003, 48313, 112047, 206097, 258563, 1
Offset: 0

Views

Author

Petros Hadjicostas, Aug 05 2020

Keywords

Comments

This is J. M. Bergot's triangular array described in A104858 with the top vertex of the triangle shifted from (1,1) to (0,0).

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   9,   1;
  1,  7,  21,  31,    1;
  1,  9,  37,  89,  121,    1;
  1, 11,  57, 183,  393,  515,    1;
  1, 13,  81, 321,  897, 1805, 2321,     1;
  1, 15, 109, 511, 1729, 4431, 8557, 10879, 1;
  ...

Crossrefs

Cf. A001003, A005408, A006318, A008288, A035011, A059993, A086616, A104858, A333090.

Programs

  • Maple
    A336858row := proc(n) option remember; local T, k, row;
row := Array(0..n, fill=1);
if n = 0 then return row fi; T := procname(n-1);
for k from 1 to n-1 do row[k] := T[k] + T[k-1] + row[k-1] od; row end:
T := (n, k) -> A336858row(n)[k]:
seq(print(seq(T(n, k), k=0..n)), n=0..8); # Peter Luschny, Aug 06 2020
  • Mathematica
    T[, 0] = 1; T[n, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, k] + T[n-1, k-1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)

Formula

T(n,k) = T(n, k-1) + T(n-1, k) + T(n-1, k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
T(n,k) = D(n,k) - Sum_{m=1..k} b(m-1)*D(n-m, k-m) - Sum_{m=0..k-1} D(n-m, k-m-1), where D(n,k) = A008288(n,k) (square array of Delannoy numbers) and b(n) = A086616(n).
T(n,1) = A005408(n-1) = 2*n - 1 for n >= 1.
T(n,2) = A059993(n-2) = 2*n^2 - 2*n - 3 for n >= 2.
T(n,n-1) = A086616(n-1) for n >= 1.
T(n,n-2) = A035011(n-1) = A006318(n-1) - 1 for n >= 2.
Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.
Bivariate o.g.f.: (1 - y - x*y*(1 + g(x*y)))/((1 - x*y)*(1 - x - y - x*y)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).
Bivariate o.g.f.: (1 - y - 2*x*y*q(x*y))/((1 - x*y)*(1 - x - y - x*y)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).
T(2*n,n) = A333090(n). - Peter Luschny, Aug 06 2020

A381943 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^2, where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 11, 60, 425, 3426, 29619, 267738, 2497889, 23866056, 232325475, 2295889266, 22971682893, 232248775669, 2368969672183, 24348849065860, 251930963865061, 2621914660411919, 27428338267887815, 288258167672381602, 3042002859317810001, 32222429872821051817
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Partial sums of A364592.
Cf. A086616, A381867, A381945.
Cf. A001764.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(n+k+1, n-k)/(4*k+1));

Formula

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

A381945 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^2, where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 3, 12, 79, 695, 6961, 74679, 837336, 9689234, 114822820, 1386402276, 16994276781, 210919650044, 2645218761934, 33470438908615, 426758782807956, 5477657372957314, 70720821402587371, 917801926609131194, 11966203939448781600, 156662012236067711036, 2058709975008385135863
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Cf. A086616, A381867, A381943.
Cf. A002293.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(n+k+1, n-k)/(5*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(n+k+1,n-k)/(5*k+1).

A336859 Mirror image of triangular array A336858.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 5, 1, 1, 31, 21, 7, 1, 1, 121, 89, 37, 9, 1, 1, 515, 393, 183, 57, 11, 1, 1, 2321, 1805, 897, 321, 81, 13, 1, 1, 10879, 8557, 4431, 1729, 511, 109, 15, 1, 1, 52465, 41585, 22149, 9161, 3001, 761, 141, 17, 1, 1, 258563, 206097, 112047, 48313, 17003, 4841, 1079, 177, 19, 1
Offset: 0

Views

Author

Petros Hadjicostas, Aug 05 2020

Keywords

Comments

This is a mirror image of A336858, which is a shifted version of J. M. Bergot's triangular array first described in A104858.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,     1;
  1,     3,    1;
  1,     9,    5,    1;
  1,    31,   21,    7,    1;
  1,   121,   89,   37,    9,   1;
  1,   515,  393,  183,   57,  11,   1;
  1,  2321, 1805,  897,  321,  81,  13,  1;
  1, 10879, 8557, 4431, 1729, 511, 109, 15, 1;
  ...

Crossrefs

Cf. A001003, A006318, A086616, A104858, A333090, A336858.

Programs

  • PARI
    A000108(n) = binomial(2*n, n)/(n+1);
A086616(n) = sum(k=0, n, binomial(n+k+1, 2*k+1) * A000108(k));
T(n, k) = if ((k==0) || (n==k), 1, if ((n<0) || (k<0), 0, if (k==1, A086616(n-1), if (n>k, T(n, k-1) - T(n-1, k-1) - T(n-1, k-2), 0))));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 08 2020

Formula

T(n,k) = A336858(n, n-k) for 0 <= k <= n.
T(n,k) = T(n, k-1) - T(n-1, k-1) - T(n-1, k-2) for 2 <= k <= n with T(n,0) = T(n,n) = 1 for n >= 0 and T(n,1) = A086616(n-1) for n >= 1.
T(2*n,n) = A333090(n).
Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.
Bivariate o.g.f.: (x*y*(1 + g(x)) + 1 - y)/((1 - x)*(1 - y + x*y + x*y^2)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).
Bivariate o.g.f.: (2*x*y*q(x) + 1 - y)/((1 - x)*(1 - y + x*y + x*y^2)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).
Previous Showing 11-16 of 16 results.

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