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A330965
Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).
10
1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 10, 20, 14, 1, 5, 14, 35, 70, 42, 1, 6, 18, 50, 126, 252, 132, 1, 7, 22, 65, 182, 462, 924, 429, 1, 8, 26, 80, 238, 672, 1716, 3432, 1430, 1, 9, 30, 95, 294, 882, 2508, 6435, 12870, 4862, 1, 10, 34, 110, 350, 1092, 3300, 9438, 24310, 48620, 16796
OFFSET
0,5
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
LINKS
FORMULA
A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).
EXAMPLE
Array begins:
====================================================
n\k | 0 1 2 3 4 5 6 7
----+-----------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 1 2 3 4 5 6 7 8 ...
2 | 2 6 10 14 18 22 26 30 ...
3 | 5 20 35 50 65 80 95 110 ...
4 | 14 70 126 182 238 294 350 406 ...
5 | 42 252 462 672 882 1092 1302 1512 ...
6 | 132 924 1716 2508 3300 4092 4884 5676 ...
7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
...
MATHEMATICA
A330965[n_, k_] := CatalanNumber[n]*(k*n + 1);
Table[A330965[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 24 2025 *)
PROG
(PARI) T(n, k)={(1 + k*n)*binomial(2*n, n)/(n+1)}
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 04 2020
STATUS
approved