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.

A337167 a(n) = 1 + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 25, 199, 1795, 17422, 177463, 1870960, 20241403, 223438852, 2506596547, 28494103183, 327507800725, 3799735202218, 44440058006593, 523388751658831, 6201937444137619, 73888034816382820, 884517283667145259, 10634234680321209373, 128347834921058404249
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 28 2021

Keywords

Comments

Binomial transform of A005159.

Links

Crossrefs

Column k=3 of A340968.
Cf. A000108, A005159, A007317, A162326, A337169, A338979.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 20}]
Table[Hypergeometric2F1[1/2, -n, 2, -12], {n, 0, 20}]
  • PARI
    {a(n) = sum(k=0, n, 3^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))} \\ Seiichi Manyama, Jan 31 2021
  • PARI
    my(N=20, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)*(1-13*x)))) \\ Seiichi Manyama, Feb 01 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + 3*x*A(x)^2.
G.f.: (1 - sqrt(1 - 12*x / (1 - x))) / (6*x).
E.g.f.: exp(7*x) * (BesselI(0,6*x) - BesselI(1,6*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Catalan(k).
a(n) = 2F1([1/2, -n], [2], -12), where 2F1 is the hypergeometric function.
D-finite with recurrence (n+1) * a(n) = 2 * (7*n-3) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1. - Seiichi Manyama, Jan 31 2021
a(n) ~ 13^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 14 2021

A340970 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(2*j,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 11, 1, 1, 7, 33, 45, 1, 1, 9, 67, 245, 195, 1, 1, 11, 113, 721, 1921, 873, 1, 1, 13, 171, 1593, 8179, 15525, 3989, 1, 1, 15, 241, 2981, 23649, 95557, 127905, 18483, 1, 1, 17, 323, 5005, 54691, 361449, 1137709, 1067925, 86515, 1
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2021

Keywords

Examples

			Square array begins:
  1,   1,     1,     1,      1,       1, ...
  1,   3,     5,     7,      9,      11, ...
  1,  11,    33,    67,    113,     171, ...
  1,  45,   245,   721,   1593,    2981, ...
  1, 195,  1921,  8179,  23649,   54691, ...
  1, 873, 15525, 95557, 361449, 1032801, ...

Links

Crossrefs

Columns k=0..3 give A000012, A026375, A084771, A340973.
Rows n=0..2 give A000012, A005408, A080859.
Main diagonal gives A340971.
Cf. A340968.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * Binomial[2*j, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *)
  • PARI
    T(n, k) = sum(j=0, n, k^j*binomial(n, j)*binomial(2*j, j));
  • PARI
    T(n, k) = polcoef((1+(2*k+1)*x+(k*x)^2)^n, n);

Formula

G.f. of column k: 1/sqrt((1 - x) * (1 - (4*k+1)*x)).
T(n,k) = [x^n] (1+(2*k+1)*x+(k*x)^2)^n.
n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (4*k+1) * (n-1) * T(n-2,k) for n > 1.
E.g.f. of column k: exp((2*k+1)*x) * BesselI(0,2*k*x). - Ilya Gutkovskiy, Feb 01 2021
From Seiichi Manyama, Aug 19 2025: (Start)
T(n,k) = (1/4)^n * Sum_{j=0..n} (4*k+1)^j * binomial(2*j,j) * binomial(2*(n-j),n-j).
T(n,k) = Sum_{j=0..n} (-k)^j * (4*k+1)^(n-j) * binomial(n,j) * binomial(2*j,j). (End)

A386387 a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k).

Original entry on oeis.org

1, 5, 41, 429, 5073, 64469, 859385, 11853949, 167763361, 2422342053, 35543185353, 528450589005, 7943934373233, 120537517728117, 1843702988611737, 28397640862311453, 440070304667718465, 6856488470912854853, 107340528355762710377, 1687682549936270584045
Offset: 0

Views

Author

Seiichi Manyama, Aug 20 2025

Keywords

Links

Crossrefs

Column k=4 of A340968.
Cf. A386389, A007317.

Programs

  • Magma
    [&+[4^k*Binomial(n,k) * Catalan(k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    Table[Sum[4^k*Binomial[n,k]*CatalanNumber[k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)
A386387[n_] := Hypergeometric2F1[1/2, -n, 2, -16]; Table[A386387[n], {n, 0, 19}]  (* Peter Luschny, Aug 27 2025 *)
  • PARI
    a(n) = sum(k=0, n, 4^k*binomial(n, k)*(2*k)!/(k!*(k+1)!));

Formula

G.f.: 2/(1 - x + sqrt((1-x) * (1-17*x))).
G.f. A(x) satisfies A(x) = 1/(1 - x) + 4*x*A(x)^2.
a(n) = 1 + 4 * Sum_{k=0..n-1} a(k) * a(n-1-k).
(n+1)*a(n) = (18*n-8)*a(n-1) - 17*(n-1)*a(n-2) for n > 1.
a(n) ~ 17^(n + 3/2) / (64*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2025
a(n) = hypergeom([1/2, -n], [2], -16). - Peter Luschny, Aug 27 2025

A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 29, 36, 1, 1, 9, 58, 185, 137, 1, 1, 11, 97, 532, 1257, 543, 1, 1, 13, 146, 1161, 5209, 8925, 2219, 1, 1, 15, 205, 2156, 14849, 53347, 65445, 9285, 1, 1, 17, 274, 3601, 34041, 198729, 564499, 491825, 39587, 1
Offset: 0

Views

Author

Seiichi Manyama, Aug 20 2025

Keywords

Examples

			Square array begins:
  1,    1,     1,      1,       1,       1,        1, ...
  1,    3,     5,      7,       9,      11,       13, ...
  1,   10,    29,     58,      97,     146,      205, ...
  1,   36,   185,    532,    1161,    2156,     3601, ...
  1,  137,  1257,   5209,   14849,   34041,    67657, ...
  1,  543,  8925,  53347,  198729,  562551,  1330693, ...
  1, 2219, 65445, 564499, 2748641, 9608811, 27053749, ...

Crossrefs

Columns k=0..4 give A000012, A002212(n+1), A127846(n+1), A386362, A386389.
Main diagonal gives A386432.
Cf. A000108, A340968.

Programs

  • PARI
    a(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*(j+1))!/((j+1)!*(j+2)!));

Formula

G.f. of column k: (1/x) * Series_Reversion( x/(1+(2*k+1)*x+(k*x)^2) ).
G.f. of column k: 2/(1 - (2*k+1)*x + sqrt((1-x) * (1-(4*k+1)*x))).
A(n,k) = (A340968(n+1,k) - A340968(n,k))/k for k > 0.
(n+2)*A(n,k) = (2*k+1)*(2*n+1)*A(n-1,k) - (4*k+1)*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * (2*k+1)^(n-2*j) * binomial(n,2*j) * Catalan(j).
Showing 1-4 of 4 results.

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