The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A182411 Triangle T(n,k) = (2*k)!*(2*n)!/(k!*n!*(k+n)!) with k=0..n, read by rows. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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). 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

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.

Last modified January 26 10:10 EST 2022. Contains 350598 sequences. (Running on oeis4.)