A068555 -id:A068555 - cp's OEIS frontend

A182073 Square array read by antidiagonals: T(n,k) = n!k! / (floor(n/2)! floor(k/2)! * floor((n+k)/2)!).

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 6, 3, 2, 3, 6, 30, 6, 6, 6, 6, 30, 20, 10, 4, 6, 4, 10, 20, 140, 20, 20, 12, 12, 20, 20, 140, 70, 35, 10, 15, 6, 15, 10, 35, 70, 630, 70, 70, 30, 30, 30, 30, 70, 70, 630, 252, 126, 28, 42, 12, 30, 12, 42, 28, 126, 252

Offset: 0

Peter Bala, Apr 10 2012

Compare with A068555 whose entries are given by (2*n)!*(2*k)!/(n!*k!*(n+k)!). See also A211226.

			As a square array
.n\k.|...0....1....2....3....4....5....6...
= = = = = = = = = = = = = = = = = = = = = =
..0..|...1....1....2....6....6...30...20...
..1..|...1....1....2....3....6...10...20...
..2..|...2....2....2....6....4...20...10...
..3..|...6....3....6....6...12...15...30...
..4..|...6....6....4...12....6...30...12...
..5..|..30...10...20...15...30...30...60...
..6..|..20...20...10...30...12...60...20...
...
Formatted as a triangle
.n\k.|...0....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = = = = =
..0..|...1
..1..|...1....1
..2..|...2....1....2
..3..|...6....2....2....6
..4..|...6....3....2....3....6
..5..|..30....6....6....6....6...30
..6..|..20...10....4....6....4...10...20
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A068555, A135573, A211226.

T[n_, k_] := n!*k!/(Floor[n/2]!*Floor[k/2]!*Floor[(n + k)/2]!);
TableForm[Table[T[n, k], {n, 0, 5}, {k, 0, 10}]]
Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* G. C. Greubel, Aug 20 2017 *)

That T(n,k) is an integer follows from the formulas:
T(2*n,2*k) = (2*n)!*(2*k)!/(n!*k!*(n+k)!) = A068555(n,k);
T(2*n,2*k+1) = (2*n)!*(2*k+1)!/(n!*k!*(n+k)!) = (2*k+1)*A068555(n,k);
T(2*n+1,2*k) = (2*n+1)!*(2*k)!/(n!*k!*(n+k)!) = (2*n+1)*A068555(n,k);
T(2*n+1,2*k+1) = (2*n+1)!*(2*k+1)!/(n!*k!*(n+k+1)!) = (2*k+1)*A135573(n,k).

A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.

Original entry on oeis.org

1, 2, 2, 6, 4, 6, 20, 10, 12, 20, 70, 28, 28, 40, 70, 252, 84, 72, 90, 140, 252, 924, 264, 198, 220, 308, 504, 924, 3432, 858, 572, 572, 728, 1092, 1848, 3432, 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870, 48620, 9724, 5304, 4420, 4760, 6120, 8976

Offset: 0

Bruno Berselli, Apr 27 2012

This is a companion to the triangle A068555.
Row sum is 2*A132310(n-1) + A000984(n) for n>0, where A000984(n) = T(n,0) = T(n,n). Also:
T(n,1)  = -A002420(n+1).
T(n,2)  =  A002421(n+2).
T(n,3)  = -A002422(n+3) = 2*A007272(n).
T(n,4)  =  A002423(n+4).
T(n,5)  = -A002424(n+5).
T(n,6)  =  A020923(n+6).
T(n,7)  = -A020925(n+7).
T(n,8)  =  A020927(n+8).
T(n,9)  = -A020929(n+9).
T(n,10) =  A020931(n+10).
T(n,11) = -A020933(n+11).

			Triangle begins:
      1;
      2,    2;
      6,    4,    6;
     20,   10,   12,   20;
     70,   28,   28,   40,   70;
    252,   84,   72,   90,  140,  252;
    924,  264,  198,  220,  308,  504,  924;
   3432,  858,  572,  572,  728, 1092, 1848,  3432;
  12870, 2860, 1716, 1560, 1820, 2520, 3960,  6864, 12870;
  48620, 9724, 5304, 4420, 4760, 6120, 8976, 14586, 25740, 48620;
  ...
Sum_{k=0..8} T(8,k) = 12870 + 2860 + 1716 + 1560 + 1820 + 2520 + 3960 + 6864 + 12870 = 2*A132310(7) + A000984(8) = 2*17085 + 12870 = 47040.

Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third edition), page 11.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 103.

Alexander Borisov, Quotient singularities, integer ratios of factorials and the Riemann Hypothesis, arXiv:math/0505167 [math.NT], 2005; International Mathematics Research Notices, Vol. 2008, Article ID rnn052, page 2 (Theorem 2).
Ira Gessel, Integer quotients of factorials and algebraic multivariable hypergeometric series, MIT Combinatorics Seminar, September 2011 (slides).
Hans-Christian Herbig and Mateus de Jesus Gonçalves, On the numerology of trigonometric polynomials, arXiv:2311.13604 [math.HO], 2023.
Kevin Limanta and Norman Wildberger, Super Catalan Numbers, Chromogeometry, and Fourier Summation over Finite Fields, arXiv:2108.10191 [math.CO], 2021. See Table 1 p. 2 where terms are shown as an array.

Cf. A000984, A002420-A020933, A068555, A132310.

[Factorial(2*k)*Factorial(2*n)/(Factorial(k)*Factorial(n)*Factorial(k+n)): k in [0..n], n in [0..9]];

Flatten[Table[Table[(2 k)! ((2 n)!/(k! n! (k + n)!)), {k, 0, n}], {n, 0, 9}]]

cp's OEIS Frontend

A182073 Square array read by antidiagonals: T(n,k) = n!k! / (floor(n/2)! floor(k/2)! * floor((n+k)/2)!).

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A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.

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cp's OEIS Frontend

A182073 Square array read by antidiagonals: T(n,k) = n!*k! / (floor(n/2)! * floor(k/2)! * floor((n+k)/2)!).

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Mathematica

Formula

A182411 Triangle T(n,k) = (2*k)!*(2*n)!/(k!*n!*(k+n)!) with k=0..n, read by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

A182073 Square array read by antidiagonals: T(n,k) = n!k! / (floor(n/2)! floor(k/2)! * floor((n+k)/2)!).

A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.