A202191 Triangle T(n,m) = coefficient of x^n in expansion of [x/(1-x-x^3)]^m = sum(n>=m, T(n,m) x^n).

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 6, 6, 4, 1, 4, 11, 13, 10, 5, 1, 6, 18, 27, 24, 15, 6, 1, 9, 30, 51, 55, 40, 21, 7, 1, 13, 50, 94, 116, 100, 62, 28, 8, 1, 19, 81, 171, 234, 231, 168, 91, 36, 9, 1, 28, 130, 303, 460, 505, 420, 266, 128, 45, 10, 1, 41, 208

Offset: 1

Author

Vladimir Kruchinin, Dec 14 2011

Keywords

• nonn
• tabl

Comments

Convolution triangle of Narayana's cows sequence A000930. - Peter Luschny, Oct 09 2022

Examples

			Triangle T(n, m) starts:
[1]  1;
[2]  1,  1;
[3]  1,  2,  1;
[4]  2,  3,  3,   1;
[5]  3,  6,  6,   4,   1;
[6]  4, 11, 13,  10,   5,  1;
[7]  6, 18, 27,  24,  15,  6,  1;
[8]  9, 30, 51,  55,  40, 21,  7,  1;
[9] 13, 50, 94, 116, 100, 62, 28,  8,  1;
.
From _R. J. Mathar_, Mar 15 2013: (Start)
The matrix inverse starts
1;
-1,1;
1,-2,1;
-2,3,-3,1;
5,-6,6,-4,1;
-11,15,-13,10,-5,1;
24,-36,33,-24,15,-6,1;
-57,84,-84,63,-40,21,-7,1;
141,-204,208,-168,110,-62,28,-8,1.
(End)
		

Crossrefs

Cf. A000930.

Programs

Maple

A202191 := proc(n,k)
    (x/(1-x-x^3))^k ;
    coeftayl(%,x=0,n) ;
end proc: # R. J. Mathar, Mar 15 2013
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> simplify(hypergeom([(2 - n)/3, (3 - n)/3, (1 - n)/3], [(2 - n)/2, (1 - n)/2], -27/4))); # Peter Luschny, Oct 09 2022

Maxima

T(n,m):=sum(binomial(k,(n-m-k)/2)*binomial(m+k-1,m-1)*((-1)^(n-m-k)+1),k,0,n-m)/2;

Formula

T(n,m)=sum(k=0..n-m, binomial(k,(n-m-k)/2)*binomial(m+k-1,m-1)*((-1)^(n-m-k)+1))/2.