A296770 -id:A296770 - cp's OEIS frontend

A050158 T(n, k) = S(2*n + 1, n, k + 1) for 0<=k<=n and n >= 0, array S as in A050157.

1, 2, 3, 5, 9, 10, 14, 28, 34, 35, 42, 90, 117, 125, 126, 132, 297, 407, 451, 461, 462, 429, 1001, 1430, 1638, 1703, 1715, 1716, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435, 4862, 11934, 18122, 21930, 23630, 24174, 24293

Offset: 0

Clark Kimberling

			Triangle starts:
                               1
                              2, 3
                            5, 9, 10
                         14, 28, 34, 35
                     42, 90, 117, 125, 126
                  132, 297, 407, 451, 461, 462
            429, 1001, 1430, 1638, 1703, 1715, 1716

T(n, 0) = A000108(n+1).
T(n, 1) = A000245(n+1).
T(n, n) = A001700(n).
T(n,n-1) = A010763(n).
Row sums are A296770.
Cf. A039598, A050157.

A050158 := (n, k) ->  binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1):
seq(seq(A050158(n,k), k=0..n), n=0..6); # Peter Luschny, Dec 22 2017

T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A039598.

T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1). Peter Luschny, Dec 22 2017

A296771 Row sums of A050157.

Original entry on oeis.org

1, 3, 13, 58, 257, 1126, 4882, 20980, 89497, 379438, 1600406, 6720748, 28117498, 117254268, 487589572, 2022568168, 8371423177, 34581780478, 142605399982, 587138954428, 2413944555742, 9911778919348, 40650232625212, 166534680737368, 681576405563722

Offset: 0

Peter Luschny, Dec 21 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..1657

Cf. A000346, A037965, A050157, A296770.

A296771 := n -> add(binomial(2*n, n) - binomial(2*n, n+k+1), k=0..n):
seq(A296771(n), n=0..24);

a[n_] := 4^n ((n - 1/2)! (2 n + 3)/(2 Sqrt[Pi] n!) - 1/2);
Table[a[n], {n, 0, 24}]

a(n) = sum(k=0, n, binomial(2*n, n) - binomial(2*n, n+k+1)) \\ Iain Fox, Dec 21 2017

a(n) = Sum_{k=0..n} (binomial(2*n, n) - binomial(2*n, n+k+1)).
a(n) = 2^(2*n-1)*(((n-1/2)!*(2*n+3))/(sqrt(Pi)*n!) - 1).
a(n) ~ 4^n*(sqrt(n/Pi) - 1/2).
a(n) = A037965(n+1) - A000346(n-1) for n >= 1.
From Robert Israel, Dec 21 2017: (Start)
a(n) = (n+3/2)*binomial(2*n,n) - 2^(2*n-1).
G.f.: (3/2-4*x)*(1-4*x)^(-3/2) - (1/2)*(1-4*x)^(-1).
64*(n+1)*(2*n+1)*a(n)-8*(2*n+3)*(5*n+4)*a(n+1)+2*(n+2)*(8*n+11)*a(n+2)-(n+3)*(n+2)*a(n+3)=0. (End)

cp's OEIS Frontend

A050158 T(n, k) = S(2*n + 1, n, k + 1) for 0<=k<=n and n >= 0, array S as in A050157.

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