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.

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A111673 Triangle, generated from A111579.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 15, 11, 4, 1, 1, 1, 52, 49, 19, 5, 1, 1, 1, 203, 257, 109, 29, 6, 1, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1, 1
Offset: 0

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Author

Gary W. Adamson, Aug 14 2005

Keywords

Comments

Columns are inverse binomial transforms of columns (k>0) of A111579.

Examples

			First few rows of the triangle are:
  1,
  1, 1,
  1, 1, 1,
  1, 2, 1, 1,
  1, 5, 3, 1, 1,
  1, 15, 11, 4, 1, 1,
  1, 52, 49, 19, 5, 1, 1,
  1, 203, 257, 109, 29, 6, 1, 1,
  1, 877, 1539, 742, 201, 41, 7, 1, 1,
  1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1,
  ...
Inverse binomial transform of column 2 of A111579 (1, 2, 5, 15, 52, 203...) = column 2 (1, 1, 2, 5, 15, 52...).

Links

Crossrefs

Cf. A111579, A008277.
For two other versions of this triangle see A241578, A241579.
Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

Extensions

More terms from N. J. A. Sloane, Apr 29 2014

A351810 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 4*x)) / (1 - 4*x)^2.

Original entry on oeis.org

1, 1, 9, 69, 565, 5305, 56929, 680685, 8902349, 126121313, 1923133433, 31379181461, 544931376229, 10024917092105, 194602995875985, 3972686705253181, 85035210652191485, 1903471938128641457, 44453001710603619369, 1080789854059236415973, 27304602412815047204501
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 19 2022

Keywords

Crossrefs

Cf. A004213, A040027, A326324, A351756, A351757, A351811, A351812.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 4 x)]/(1 - 4 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

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

A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 2)^n / (4^k * k!).

Original entry on oeis.org

1, 3, 13, 79, 601, 5339, 53861, 607527, 7560625, 102637235, 1506225085, 23726435583, 398852249097, 7120170905995, 134408217821205, 2673140092099543, 55832167947587425, 1221199519275467107, 27902127744298836845, 664446811342185649583, 16457968670922936733113, 423242969435491221774907
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 22 2022

Keywords

Crossrefs

Cf. A003576, A004213, A285064, A355164, A355167.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[2 x + (Exp[4 x] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n, k] 2^(n + k) BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]

Formula

E.g.f.: exp(2*x + (exp(4*x) - 1) / 4).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A004213(k).
a(n) ~ 2^(2*n+1) * n^(n + 1/2) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2022
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