Original entry on oeis.org
1, -4, 4, 8, -32, 32, 64, -256, 256, 512, -2048, 2048, 4096, -16384, 16384, 32768, -131072, 131072, 262144, -1048576, 1048576, 2097152, -8388608, 8388608, 16777216, -67108864, 67108864, 134217728, -536870912, 536870912
Offset: 0
Apart from signs, essentially the same as
A096252.
A182411
Triangle T(n,k) = (2*k)!*(2*n)!/(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
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.
-
[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}]]
Comments