A060086 Convolution triangle A059594 with extra first column.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 3, 14, 19, 14, 5, 1, 0, 4, 20, 39, 36, 20, 6, 1, 0, 4, 30, 69, 85, 60, 27, 7, 1, 0, 5, 40, 119, 176, 160, 92, 35, 8, 1, 0, 5, 55, 189, 344, 376, 273, 133, 44, 9

Offset: 0

Author

Wolfdieter Lang, Apr 06 2001

Keywords

• nonn
• easy
• tabl

Comments

Riordan array (1, x/((1+x)*(1-x)^2)). - Philippe Deléham, Feb 24 2012

Triangle, read by rows, given by (0, 1, 1, -2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 24 2012

Examples

			{1}; {0,1}; {0,1,1}; {0,2,2,1}; ...
Triangle begins :
1
0, 1
0, 1, 1
0, 2, 2, 1
0, 2, 5, 3, 1
0, 3, 8, 9, 4, 1
0, 3, 14, 19, 14, 5, 1
		

Crossrefs

Cf. A059594,

Programs

Mathematica

t[0, 0] = 1; t[, 0] = 0; t[n, m_] := Sum[ Sum[ Binomial[j, 2*j-3*k-m+n]*(-1)^(j-k)*Binomial[k, j], {j, 0, k}]*Binomial[m+k-1, m-1], {k, 0, n-m}]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

Formula

G.f.for column m >= 0: (x/((1-x^2)*(1-x)))^m.

T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) - T(n-3,k) with T(n,0) = 0^n. - Philippe Deléham, Feb 24 2012

G.f.: (1-x-x^2+x^3)/(1-x-x^2+x^3-y*x). - Philippe Deléham, Feb 24 2012

Sum_{k, 0<=k<=n} T(n,k)*2^k = A181301(n). - Philippe Deléham, Feb 24 2012