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A111910 Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0). 6
1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 84, 30, 1, 1, 55, 330, 330, 55, 1, 1, 91, 1001, 2145, 1001, 91, 1, 1, 140, 2548, 10010, 10010, 2548, 140, 1, 1, 204, 5712, 37128, 68068, 37128, 5712, 204, 1, 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

S(n,n) = A111911(n).

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).

G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin. 2 (1981), 55-60.

Anthony J. Wood, Richard A. Blythe, and Martin R. Evans, Combinatorial mappings of exclusion processes, arXiv:1908.00942 [cond-mat.stat-mech], 2019.

FORMULA

From Peter Bala, Oct 13 2011: (Start)

Define a(n) = n!*(n+1/2)!*(n+1)!/(1/2)!.

S(n,k) = a(n+k)/(a(n)*a(k)) gives the sequence as a square array while T(n,k) = a(n)/(a(n-k)*a(k)) gives the sequence as a triangle.

S(n-1,k)*S(n,k+1)*S(n+1,k-1) = S(n-1,k+1)*S(n,k-1)*S(n+1,k). Cf. A091044.

(End)

From G. C. Greubel, Feb 12 2021: (Start)

As a number triangle:

T(n, k) = binomial(n+1, k)*binomial(2*n+1, 2*k)/((k+1)*(2*k+1)).

T(n, k) = binomial(2*n+1, 2*k)/((2*k+1)*binomial(n, k)) * A001263(n+1, k+1). (End)

EXAMPLE

Array S(n,k) in rectangular form (n, k >= 0):

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

  1,  5,   14,    30,   55,      91,     140,     204,      285, ...

  1, 14,   84,   330,  1001,   2548,    5712,   11628,    21945, ...

  1, 30,  330,  2145, 10010,  37128,  116280,  319770,   793155, ...

  1, 55, 1001, 10010, 68068, 352716, 1492260, 5393454, 17185025, ...

  ...

Array T(n,k) in triangular form (n >= 0 and 0 <= k <= n):

  1,

  1,  1,

  1,  5,    1,

  1, 14,   14,   1,

  1, 30,   84,  30,     1,

  1, 55,  330, 330,    55,  1,

  1, 91, 1001, 2145, 1001, 91, 1,

  ...

MAPLE

a:=(p, q)->(p+q+1)!*(2*p+2*q+1)!/(p+1)!/(2*p+1)!/(q+1)!/(2*q+1)!: for n from 0 to 10 do seq(a(j, n-j), j=0..n) od; # yields sequence in triangular form

MATHEMATICA

Table[(# + q + 1)! (2 # + 2 q + 1)!/((# + 1)! (2 # + 1)! (q + 1)! (2 q + 1)!) &[r - q], {r, 0, 9}, {q, 0, r}] // Flatten (* Michael De Vlieger, Oct 21 2019 *)

Table[Binomial[n+1, k]*Binomial[2*n+1, 2*k]/((k+1)*(2*k+1)), {n, 0, 12}, {k, 0,

n}]//Flatten (* G. C. Greubel, Feb 12 2021 *)

PROG

(Sage)

def A111910(n, k): return binomial(n+1, k)*binomial(2*n+1, 2*k)/((k+1)*(2*k+1))

flatten([[A111910(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 12 2021

(Magma)

T:= func< n, k | Binomial(n+1, k)*Binomial(2*n+1, 2*k)/((k+1)*(2*k+1)) >;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021

CROSSREFS

Cf. A091044, A111911, A196148 (row sums of triangle).

Cf. A001263.

Sequence in context: A119725 A239279 A278880 * A181143 A144438 A157207

Adjacent sequences:  A111907 A111908 A111909 * A111911 A111912 A111913

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Aug 19 2005

EXTENSIONS

Example section edited by Petros Hadjicostas, Sep 03 2019

STATUS

approved

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Last modified August 4 14:45 EDT 2021. Contains 346447 sequences. (Running on oeis4.)