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

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

0,2

Comments

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).

References

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.

Links

Table of n, a(n) for n=0..51.

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A081111 A092686 A249796 * A067804 A074911 A174222

Adjacent sequences: A182408 A182409 A182410 * A182412 A182413 A182414

Keyword

nonn,tabl,look,easy

Author

Bruno Berselli, Apr 27 2012

Status

approved