A188285 Riordan matrix ( (1-2x)/(1-2x-x^2), (x-2x^2)/(1-2x-x^2) ).

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 4, 3, 0, 1, 12, 11, 6, 4, 0, 1, 29, 28, 18, 8, 5, 0, 1, 70, 72, 48, 26, 10, 6, 0, 1, 169, 184, 130, 72, 35, 12, 7, 0, 1, 408, 469, 348, 204, 100, 45, 14, 8, 0, 1, 985, 1192, 927, 568, 295, 132, 56, 16, 9, 0, 1, 2378, 3022, 2456, 1571, 850, 404, 168, 68, 18, 10, 0, 1, 5741, 7644, 6477, 4312, 2430, 1200, 532, 208, 81, 20, 11, 0, 1

Offset: 0

Author

Emanuele Munarini, Mar 26 2011

Keywords

• nonn
• tabl
• easy

Comments

T(n,k) is the number of Dyck paths of height at most 3 with length 2n and k hills.

Row sum = F_(2n-1) Fibonacci number.

T is the convolution triangle of |A215936|. - Peter Luschny, Oct 19 2022

Examples

			Triangle begins:
1
0, 1
1, 0, 1
2, 2, 0, 1
5, 4, 3, 0, 1
12, 11, 6, 4, 0, 1
29, 28, 18, 8, 5, 0, 1
70, 72, 48, 26, 10, 6, 0, 1
169, 184, 130, 72, 35, 12, 7, 0, 1
408, 469, 348, 204, 100, 45, 14, 8, 0, 1
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..495

Programs

Maple

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> (-1)^(n+1)*A215936(n)); # Peter Luschny, Oct 19 2022

Mathematica

Flatten[Table[Sum[Pochhammer[i,n-k-2i]/(n-k-2i)!Binomial[i+k,k]2^(n-k-2i),{i,0,(n-k)/2}],{n,0,12},{k,0,n}],1]

Maxima

create_list(sum(pochhammer(i,n-k-2*i)/(n-k-2*i)!*binomial(i+k,k)*2^(n-k-2*i),i,0,(n-k)/2),n,0,12,k,0,n);

Formula

T(n,k) = sum(M(i,n-k-2i)*Binomial(i+k,k)*2^{n-k-2i},i=0..floor((n-k)/2)), where M(n,k)=n(n+1)(n+2)...(n+k-1)/k!.

Recurrence: T(n+2,k+1) = 2 T(n+1,k+1) + T(n+1,k) + T(n,k+1) - 2 T(n,k)