A137551 -id:A137551 - cp's OEIS frontend

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 - 2x - 2x^2/(1 - 3x - 3x^2/(1 - 4x - 4x^2/(1 - ...))))).

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

Ilya Gutkovskiy, Sep 25 2017

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

			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,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

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).

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

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

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.

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

Philippe Deléham, Jan 29 2012

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.

			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;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

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

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

Sum_{k=0..n} T(n,k) = A137551(n), n>0.

A129247 Invert transform of the Bell numbers.

Original entry on oeis.org

1, 1, 3, 10, 36, 138, 560, 2402, 10898, 52392, 267394, 1450790, 8371220, 51327178, 333759746, 2295276480, 16639104002, 126718172670, 1010487248556, 8411744415418, 72899055533482, 656136245454232, 6120474697035762

Offset: 0

Thomas Wieder, May 10 2008

nonn

The following definition of the invert transform appears in [M. Bernstein & N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra and its Applications, 226-228 (1995), 57-72]: "b_n is the number of ordered arrangements of postage stamps of total value n that can be formed if we have a_i types of stamps of value i, i >= 1."
Hankel transform is A000178. - Paul Barry, Jan 08 2009
Equals INVERT transform of the Bell sequence starting with offset 1: (1, 2, 5, ...), while A137551 = INVERT transform of the Bell sequence starting with offset 0: (1, 1, 2, 5, 15, 52, ...). - Gary W. Adamson, May 24 2009

			We have Bell(i) types of an integer i with i=1,2,...,n, where Bell(i) is the i-th Bell number.
We write i_j for integer i of type j.
a(2)=3 because of the 3 ordered arrangements
  {1_1,1_1}
  {2_1}, {2_2}.
a(3)=10 because of the 10 ordered arrangements
  {1_1,1_1,1_1},
  {1_1,2_1}, {2_1,1_1},
  {1_1,2_2}, {2_2,1_1}
  {3_1}, {3_2}, {3_3}, {3_4}, {3_5}.

Alois P. Heinz, Table of n, a(n) for n = 0..576
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
T. Mansour and M. Shattuck, A statistic on n-color compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) 124(2) (2014), pp. 127-140.
N. J. A. Sloane, Transforms

Cf. A000110, A055887, A083355.

A129247 := proc(n) option remember ; local i ; if n <= 1 then 1 ; else add(combinat[bell](i)*procname(n-i),i=1..n) ; fi ; end: for n from 0 to 40 do printf("%d,",A129247(n)) ; od: # R. J. Mathar, Aug 25 2008

a[0] = 1; a[n_] := a[n] = Sum[BellB[i]*a[n - i], {i, 1, n}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 09 2017 *)

a(n) = Sum_{i=1..n} Bell(i)*a(n-i).
G.f.: 1/(U(0) - 2*x) where U(k) = 1 - x*(k+1)/(1 - x/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 12 2012
G.f.: 1/( Q(0) - 2*x ) where Q(k) = 1 + x/(x*k - 1 )/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Feb 23 2013
G.f.: 1/(Q(0) - x), where Q(k) = 1 - x - x/(1 - x*(2*k+1)/(1 - x - x/(1 - 2*x*(k+1)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, May 12 2013

Extended by R. J. Mathar, Aug 25 2008

a(0)=1 prepended by Alois P. Heinz, Sep 22 2017

A335849 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Bell(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 3, 14, 87, 675, 6282, 68201, 846183, 11811048, 183176577, 3124958179, 58157682072, 1172551946395, 25459025908899, 592263131497942, 14696581853565723, 387477880784385143, 10816856730117090114, 318739828787430822853, 9886623306152849028771

Offset: 0

Ilya Gutkovskiy, Jun 26 2020

nonn

Cf. A000110, A004122, A083355, A129247, A137551.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[1]/(Exp[1] + ExpIntegralEi[1] - ExpIntegralEi[Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(1) / (exp(1) + Ei(1) - Ei(exp(x))), where Ei() is the exponential integral.

cp's OEIS Frontend

A160701 Duplicate of A137551.

Views

Author

Keywords

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 - ...))))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A129247 Invert transform of the Bell numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A335849 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Bell(k-1) * a(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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 - 2x - 2x^2/(1 - 3x - 3x^2/(1 - 4x - 4x^2/(1 - ...))))).