A131084 A129686 * A007318. Riordan triangle (1+x, x/(1-x)).

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 2, 5, 4, 1, 0, 2, 7, 9, 5, 1, 0, 2, 9, 16, 14, 6, 1, 0, 2, 11, 25, 30, 20, 7, 1, 0, 2, 13, 36, 55, 50, 27, 8, 1, 0, 2, 15, 49, 91, 105, 77, 35, 9, 1

Offset

1,5

Comments

Row sums = A098011 starting (1, 2, 3, 6, 12, 24, 48, ...). A131085 = A007318 * A129686

Riordan array (1+x, x/(1-x)). - Philippe Deléham, Mar 02 2012

Links

Table of n, a(n) for n=1..55.

Formula

A129686(signed): (1,1,1,...) in the main diagonal and (-1,-1,-1, ...) in the subsubdiagonal); * A007318, Pascal's triangle; as infinite lower triangular matrices.

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2*x + 3*x^2/2! + x^3/3!) = 2*x + 7*x^2/2! + 16*x^3/3! + 30*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

G.f. column k: (1+x)*(x/(1-x))^k, k >= 0. (Riordan property). - Wolfdieter Lang, Jan 06 2015

T(n, 0) = 1 if n=0 or n=1 else 0; T(n, k) = binomial(n-1,k-1) + binomial(n-2,k-1)*[n-1 >= k] if n >= k >= 1, where [S] = 1 if S is true, else 0, and T(n, k) = 0 if n < k. - Wolfdieter Lang, Jan 08 2015

Example

The triangle T(n, k) begins:
n\k 0  1  2  3   4   5   6   7  8  9 10 ...
0:  1
1:  1  1
2:  0  2  1
3:  0  2  3  1
4:  0  2  5  4   1
5:  0  2  7  9   5   1
6:  0  2  9 16  14   6   1
7:  0  2 11 25  30  20   7   1
8:  0  2 13 36  55  50  27   8  1
9:  0  2 15 49  91 105  77  35  9  1
10: 0  2 17 64 140 196 182 112 44 10  1
... Reformatted. - Wolfdieter Lang, Jan 06 2015

Crossrefs

Cf. A007318, A129686, A098011, A131085.

Cf. A029653, A131084, A208510.

Sequence in context: A104041 A104402 A261897 * A143067 A219605 A356027

Adjacent sequences: A131081 A131082 A131083 * A131085 A131086 A131087

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 14 2007

Extensions

Edited: Added Riordan property (see Philippe Deléham comment) in name. - Wolfdieter Lang, Jan 06 2015

Status

approved