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A257488
Triangle, read by rows, T(n,k) = k*Sum_{i=0..n-k} C(2*i+2*k,i)*C(n-i-1,k-1)/(i+k) for 1 <= k <= n.
0
1, 3, 1, 8, 6, 1, 22, 25, 9, 1, 64, 92, 51, 12, 1, 196, 324, 237, 86, 15, 1, 625, 1128, 996, 484, 130, 18, 1, 2055, 3934, 3966, 2377, 860, 183, 21, 1, 6917, 13812, 15335, 10744, 4845, 1392, 245, 24, 1, 23713, 48884, 58359, 46068, 24603, 8859, 2107, 316, 27, 1
OFFSET
1,2
FORMULA
G.f.: 1/(1-(C(x)-1)/(1-x)*y)-1, where C(x) is g.f. of Catalan numbers (A000108).
T(n,n-1) = 3*(n-1) for n > 1. - Derek Orr, Apr 27 2015
T(n,n-2) = A062728(n-2) for n > 2. - Derek Orr, Apr 27 2015
T(n,1) = A014138(n). - Derek Orr, Apr 27 2015
EXAMPLE
Triangle starts:
1;
3, 1;
8, 6, 1;
22, 25, 9, 1;
64, 92, 51, 12, 1;
MATHEMATICA
Flatten@ Table[k Sum[Binomial[2 i + 2 k, i] Binomial[n - i - 1, k - 1]/(i + k), {i, 0, n - k}], {n, 10}, {k, n}] (* Michael De Vlieger, Apr 27 2015 *)
PROG
(Maxima)
T(n, k):=k*sum((binomial(2*i+2*k, i)*binomial(n-i-1, k-1))/(i+k), i, 0, n-k);
(PARI) T(n, k)=k*sum(i=0, n-k, (binomial(2*i+2*k, i)*binomial(n-i-1, k-1))/(i+k))
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ Derek Orr, Apr 27 2015
CROSSREFS
Cf. A014138.
Sequence in context: A123965 A125662 A124025 * A286416 A340672 A005295
KEYWORD
nonn,tabl,easy
AUTHOR
Vladimir Kruchinin, Apr 26 2015
STATUS
approved