A292871 -id:A292871 - 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.

A309955 a(n) = [x^n] (1 + p(x))^n, where p(x) is the g.f. of A000040.

Original entry on oeis.org

1, 2, 10, 59, 362, 2287, 14707, 95762, 629386, 4166627, 27743445, 185602188, 1246543559, 8399791922, 56762121398, 384513835219, 2610322687850, 17753944125159, 120954505004605, 825274753259894, 5638438272353597, 38569743775323134, 264127692090124488

Offset: 0

Alois P. Heinz, Aug 24 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1185

Cf. A000040, A008485, A008578, A025174, A292871, A309950, A340990.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, ithprime(n),
      (h-> add(b(j, h)*b(n-j, i-h), j=0..n))(iquo(i, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..31);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Prime[n],
     Function[h, Sum[b[j, h]*b[n-j, i-h], {j, 0, n}]][Quotient[i, 2]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

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 - ...))))).

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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.

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A309955 a(n) = [x^n] (1 + p(x))^n, where p(x) is the g.f. of A000040.

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

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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

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Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A309955 a(n) = [x^n] (1 + p(x))^n, where p(x) is the g.f. of A000040.

Views

Author

Keywords

Links

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Programs

Maple

Mathematica

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 - ...))))).