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.

A014322 Convolution of Bell numbers with themselves.

Original entry on oeis.org

1, 2, 5, 14, 44, 154, 595, 2518, 11591, 57672, 308368, 1762500, 10716321, 69011130, 468856113, 3348695194, 25064539520, 196052415230, 1598543907843, 13556379105766, 119332020447219, 1088376385244908, 10268343703117892, 100063762955374568, 1005822726810785809
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Equals row sums of triangle A144155. - Gary W. Adamson, Sep 12 2008

Links

Crossrefs

Cf. A000110, A014323, A014325, A144155.
Column k=2 of A292870.

Programs

  • Magma
    A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
[A014322(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023
  • Maple
    with(combinat):
a:= n-> add(bell(i)*bell(n-i), i=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 13 2014
  • Mathematica
    a[n_]:= Sum[BellB[k]*BellB[n-k], {k,0,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Jan 17 2016 *)
  • SageMath
    def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
[A014322(n) for n in range(41)] # G. C. Greubel, Jan 08 2023

Formula

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^2, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017
G.f.: ( Sum_{j>=0} A000110(j)*x^j )^2. - G. C. Greubel, Jan 08 2023

A292870 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 15, 0, 1, 5, 14, 28, 44, 52, 0, 1, 6, 20, 48, 93, 154, 203, 0, 1, 7, 27, 75, 169, 333, 595, 877, 0, 1, 8, 35, 110, 280, 624, 1289, 2518, 4140, 0, 1, 9, 44, 154, 435, 1071, 2442, 5394, 11591, 21147, 0, 1, 10, 54, 208, 644, 1728, 4265, 10188, 24366, 57672, 115975, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2017

Keywords

Comments

A(n,k) is the n-th term of the k-fold convolution of Bell numbers with themselves. - Alois P. Heinz, Feb 12 2019

Examples

			G.f. of column k: A_k(x) = 1 + k*x + k*(k + 3)*x^2/2 + k*(k^2 + 9*k + 20)*x^3/6 + k*(k^3 + 18*k^2 + 107*k + 234)*x^4/24 + k*(k^4 + 30*k^3 + 335*k^2 + 1770*k + 4104)*x^5/120 + ...
Square array begins:
  1,   1,    1,    1,    1,     1,  ...
  0,   1,    2,    3,    4,     5,  ...
  0,   2,    5,    9,   14,    20,  ...
  0,   5,   14,   28,   48,    75,  ...
  0,  15,   44,   93,  169,   280,  ...
  0,  52,  154,  333,  624,  1071,  ...

Links

Crossrefs

Columns k=0-4 give A000007, A000110, A014322, A014323, A014325.
Rows n=0-3 give A000012, A001477, A000096, A005586.
Antidiagonal sums give A137551.
Main diagonal gives A292871.
Cf. A205574 (another version).

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
     `if`(k=1, add(A(n-j, k)*binomial(n-1, j-1), j=1..n),
     (h-> add(A(j, h)*A(n-j, k-h), j=0..n))(iquo(k,2)))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 31 2018
  • Mathematica
    Table[Function[k, SeriesCoefficient[1/(1 - x + ContinuedFractionK[-i x^2, 1 - (i + 1) x, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

G.f. of column k: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^k, a continued fraction.

A014323 Three-fold convolution of Bell numbers with themselves.

Original entry on oeis.org

1, 3, 9, 28, 93, 333, 1289, 5394, 24366, 118526, 618924, 3456942, 20573391, 129951231, 867877107, 6106194478, 45109290477, 348836705235, 2816093142803, 23673989688810, 206794355179656, 1873232870155036, 17565534522745008, 170237112831874188
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000110, A014322, A014325.
Column k=3 of A292870.

Programs

Formula

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^3, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017
From G. C. Greubel, Jan 08 2023: (Start)
a(n) = Sum_{j=0..n} A000110(j)*A014322(n-j).
G.f.: ( Sum_{j>=0} A000110(j)*x^j )^3. (End)

A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 15, 14, 9, 4, 1, 0, 52, 44, 28, 14, 5, 1, 0, 203, 154, 93, 48, 20, 6, 1, 0, 877, 595, 333, 169, 75, 27, 7, 1, 0, 4140, 2518, 1289, 624, 280, 110, 35, 8, 1, 0, 21147, 11591, 5394, 2442, 1071, 435, 154, 44, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 29 2012

Keywords

Comments

Bell convolution triangle ; g.f. for column k : (x*B(x))^k with B(x) g.f. for A000110 (Bell numbers).
Riordan array (1, x*B(x)), when B(x) the g.f. of A000110.
Row sums are in A137551.

Examples

			Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,  1;
  0,   5,   5,  3,  1;
  0,  15,  14,  9,  4,  1;
  0,  52,  44, 28, 14,  5, 1;
  0, 203, 154, 93, 48, 20, 6, 1;
  ...

Links

Crossrefs

Cf. Columns : A000007, A000110, A014322, A014323, A014325 ; Diagonals : A000012, A001477, A000096, A005586.
Another version: A292870.
T(2n,n) gives: A292871.

Programs

  • Maple
    # Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-bell(n-1)); # Peter Luschny, Oct 19 2022

Formula

Sum_{k=0..n} T(n,k) = A137551(n), n>0.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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