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

0,8

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

Links

Table of n, a(n) for n=0..64.

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

Example

{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

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 *)

Crossrefs

Cf. A059594,

Sequence in context: A326280 A350310 A280817 * A308680 A177975 A340995

Adjacent sequences: A060083 A060084 A060085 * A060087 A060088 A060089

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Apr 06 2001

Status

approved