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

0,7

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

Links

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

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)

Example

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

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]

Prog

(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);

Crossrefs

Sequence in context: A276727 A267617 A158852 * A273899 A102404 A089246

Adjacent sequences: A188282 A188283 A188284 * A188286 A188287 A188288

Keyword

nonn,tabl,easy

Author

Emanuele Munarini, Mar 26 2011

Status

approved