A290777 -id:A290777

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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 - ...)))))).

+10
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, 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.

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Aug 09 2017

STATUS

approved

A090182

Triangle T(n,k), 0 <= k <= n, composed of k-Catalan numbers.

+10
13

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 17, 4, 1, 1, 1, 42, 171, 43, 5, 1, 1, 1, 132, 3113, 1252, 89, 6, 1, 1, 1, 429, 106419, 104098, 5885, 161, 7, 1, 1, 1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1, 1, 4862, 915028347, 18649337311, 1558435125, 12833546, 57799, 407, 9, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Alois P. Heinz, Rows n = 0..55, flattened

Lun Lv, Zhihong Liu, Some Identities Related to Restricted Lattice Paths, 2016 9th International Symposium on Computational Intelligence and Design (ISCID), pp. 338-340.

EXAMPLE

Triangle begins:

1, 1;

1, 1, 1;

1, 2, 1, 1;

1, 5, 3, 1, 1;

1, 14, 17, 4, 1, 1;

1, 42, 171, 43, 5, 1, 1;

1, 132, 3113, 1252, 89, 6, 1, 1;

1, 429, 106419, 104098, 5885, 161, 7, 1, 1;

1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1;

This sequence formatted as a square array:

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

1, 1, 2, 5, 14, 42, 132, 429, ...

1, 1, 3, 17, 171, 3113, 106419, 7035649, ...

1, 1, 4, 43, 1252, 104098, 25511272, 18649337311, ...

1, 1, 5, 89, 5885, 1518897, 1558435125, 6386478643785, ...

1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, ...

MAPLE

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

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

end:

seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Aug 10 2017

MATHEMATICA

nmax = 10; col[k_] := col[k] = Module[{A}, A[_] = 0; Do[A[x_] = Normal[1/(1 - x*A[k*x]) + O[x]^(nmax-k+1)], {nmax-k+1}]; CoefficientList[A[x], x]];

T[n_, k_] := col[k][[n-k+1]];

Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2019, using g.f. given for column sequences *)

CROSSREFS

The column sequences (without leading zeros) are A000012, A000108 (Catalan), A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096 for k=0..11.

T(2n,n) gives A290777.

Cf. A290759.

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jan 20 2004, Oct 16 2008

STATUS

approved

A290786

a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -n.

+10
3

1, 1, -1, -23, 3429, 8425506, -412878084725, -497641562809372379, 17436260499054618815283977, 20503694883570579788445502041773422, -917439693541287252616828116888122637934368489, -1746281566732870051764961051797990328294109372786185933382

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..36

J. Fürlinger and J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.

Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.

FORMULA

a(n) = [x^n] 1/(1-x/(1+n*x/(1-n^2*x/(1+n^3*x/(1-n^4*x/(1+ ... )))))).

a(n) = A290789(n,n).

MAPLE

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

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

end:

a:= n-> b(n$2):

seq(a(n), n=0..12);

MATHEMATICA

b[n_, k_]:=b[n, k]=If[n==0, 1, Sum[b[j, k] b[n - j - 1, k] (-k)^j, {j, 0, n - 1}]]; Table[b[n, n], {n, 0, 15}] (* Indranil Ghosh, Aug 10 2017 *)

PROG

(Python)

from sympy.core.cache import cacheit

@cacheit

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

def a(n): return b(n, n)

print([a(n) for n in range(16)]) # Indranil Ghosh, Aug 10 2017

CROSSREFS

Main diagonal of A290789.

Cf. A290777.

KEYWORD

sign

AUTHOR

Alois P. Heinz, Aug 10 2017

STATUS

approved

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