login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 - ...))))). 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

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

LINKS

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

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.

EXAMPLE

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,  ...

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

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).

Sequence in context: A297321 A277938 A130020 * A306704 A091063 A246935

Adjacent sequences:  A292867 A292868 A292869 * A292871 A292872 A292873

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Sep 25 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 12:30 EST 2019. Contains 329895 sequences. (Running on oeis4.)