login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))). 14
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 17, 14, 1, 1, 1, 5, 43, 171, 42, 1, 1, 1, 6, 89, 1252, 3113, 132, 1, 1, 1, 7, 161, 5885, 104098, 106419, 429, 1, 1, 1, 8, 265, 20466, 1518897, 25511272, 7035649, 1430, 1, 1, 1, 9, 407, 57799, 12833546, 1558435125, 18649337311, 915028347, 4862, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

This is the transpose of the array in A090182.

LINKS

Seiichi Manyama, Antidiagonals n = 0..55, flattened

FORMULA

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

EXAMPLE

G.f. of column k: A_k(x) = 1 + x + (k + 1)*x^2 + (k^3 + k^2 + 2*k + 1)*x^3 + (k^6 + k^5 + 2*k^4 + 3*k^3 + 3*k^2 + 3*k + 1)*x^4 + ...

Square array begins:

  1,   1,     1,       1,        1,         1,  ...

  1,   1,     1,       1,        1,         1,  ...

  1,   2,     3,       4,        5,         6,  ...

  1,   5,    17,      43,       89,       161,  ...

  1,  14,   171,    1252,     5885,     20466,  ...

  1,  42,  3113,  104098,  1518897,  12833546,  ...

MAPLE

A:= proc(n, k) option remember; `if`(n=0, 1, add(

      A(j, k)*A(n-j-1, k)*k^j, j=0..n-1))

    end:

seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 10 2017

MATHEMATICA

Table[Function[k, SeriesCoefficient[1/(1 - x/(1 + ContinuedFractionK[-k^i x, 1, {i, 1, n}])), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

PROG

(Python)

from sympy.core.cache import cacheit

@cacheit

def A(n, k): return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*k**j for j in range(n))

for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 10 2017, after Maple code

CROSSREFS

Columns k=0-11 give: A000012, A000108, A015083, A015084, A015085, A015086, A015089,  A015091, A015092, A015093, A015095, A015096.

Main diagonal gives A290777.

Cf. A090182, A290789.

Sequence in context: A305962 A144150 A124560 * A306245 A275043 A227061

Adjacent sequences:  A290756 A290757 A290758 * A290760 A290761 A290762

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Aug 09 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 August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)