A247492 Triangle read by rows: T(n, k) = binomial(k-1, n-k)*(n+1)/(n+1-k), 0 <= k <= n.

1, -1, 2, 1, 0, 3, -1, 0, 2, 4, 1, 0, 0, 5, 5, -1, 0, 0, 2, 9, 6, 1, 0, 0, 0, 7, 14, 7, -1, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 9, 30, 27, 9, -1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, -1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12

Offset

0,3

Links

Table of n, a(n) for n=0..77.

Formula

Sum_{k = 0..n} T(n, k) = A001350(n+1).

G.f.: (x^2*y + 1)/((x^4 + 2*x^3 + x^2)*y^2 + (-x^3 - 3*x^2 - 2*x)*y + x + 1). Or: (x^2*y + 1)/((x + 1)*(x*y - 1)*(x^2*y + x*y - 1)). - Vladimir Kruchinin, Oct 23 2021

Example

[0]  1;
[1] -1, 2;
[2]  1, 0, 3;
[3] -1, 0, 2, 4;
[4]  1, 0, 0, 5, 5;
[5] -1, 0, 0, 2, 9, 6;
[6]  1, 0, 0, 0, 7, 14, 7;
.
Taylor series: 1 + x*(2*y - 1) + x^2*(3*y^2 + 1) + x^3*(4*y^3 + 2*y^2 - 1) + x^4*(5*y^4 + 5*y^3 + 1) + O(x^5).

Maple

T := (n, k) -> (n+1)*binomial(k-1, n-k)/(n+1-k);
for n from 0 to 11 do seq(T(n, k), k=0..n) od;

Crossrefs

Cf. A001350 (row sums), A098599, A100218.

Sequence in context: A376789 A127013 A117362 * A113214 A280494 A168016

Adjacent sequences: A247489 A247490 A247491 * A247493 A247494 A247495

Keyword

sign,tabl

Author

Peter Luschny, Oct 01 2014

Status

approved