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 A238761 Subtriangle of the generalized ballot numbers, T(n,k) = A238762(2*k-1,2*n-1), 1<=k<=n, read by rows. 2
 1, 2, 3, 3, 8, 10, 4, 15, 30, 35, 5, 24, 63, 112, 126, 6, 35, 112, 252, 420, 462, 7, 48, 180, 480, 990, 1584, 1716, 8, 63, 270, 825, 1980, 3861, 6006, 6435, 9, 80, 385, 1320, 3575, 8008, 15015, 22880, 24310, 10, 99, 528, 2002, 6006, 15015, 32032, 58344, 87516, 92378 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA T(n,n) = A001700(n-1). T(n,n-1) = A162551(n-1). EXAMPLE [n\k 1   2    3    4    5    6     7 ] [1]  1, [2]  2,  3, [3]  3,  8,  10, [4]  4, 15,  30,  35, [5]  5, 24,  63, 112, 126, [6]  6, 35, 112, 252, 420,  462, [7]  7, 48, 180, 480, 990, 1584, 1716. MAPLE binom2 := proc(n, k) local h;    h := n -> (n-((1-(-1)^n)/2))/2;    n!/(h(n-k)!*h(n+k)!) end: A238761 := (n, k) -> binom2(n+k, n-k+1)*(n-k+1)/(n+k): seq(print(seq(A238761(n, k), k=1..n)), n=1..7); MATHEMATICA h[n_] := (n - ((1 - (-1)^n)/2))/2; binom2[n_, k_] := n!/(h[n-k]! h[n+k]!); T[n_, k_] := binom2[n+k, n-k+1] (n-k+1)/(n+k); Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2019, from Maple *) PROG (Sage) @CachedFunction def ballot(p, q):     if p == 0 and q == 0: return 1     if p < 0 or p > q: return 0     S = ballot(p-2, q) + ballot(p, q-2)     if q % 2 == 1: S += ballot(p-1, q-1)     return S A238761 = lambda n, k: ballot(2*k-1, 2*n-1) for n in (1..7): [A238761(n, k) for k in (1..n)] CROSSREFS Row sums are A002054. Cf. A001700, A009766. Sequence in context: A161136 A258163 A108381 * A261469 A292498 A108692 Adjacent sequences:  A238758 A238759 A238760 * A238762 A238763 A238764 KEYWORD nonn,tabl AUTHOR Peter Luschny, Mar 05 2014 STATUS approved

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Last modified September 28 02:44 EDT 2020. Contains 337392 sequences. (Running on oeis4.)