A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).

1, 1, 1, 2, 3, 1, 4, 7, 5, 1, 7, 17, 16, 7, 1, 13, 38, 46, 29, 9, 1, 24, 82, 122, 99, 46, 11, 1, 44, 174, 304, 303, 184, 67, 13, 1, 81, 362, 728, 857, 641, 309, 92, 15, 1, 149, 743, 1690, 2291, 2031, 1212, 482, 121, 17, 1, 274, 1509, 3827, 5869, 6004, 4260, 2108, 711, 154, 19, 1

Offset

0,4

Comments

Row sums are A077936, diagonal sums are A077946

Links

Table of n, a(n) for n=0..65.

Formula

T(n,k) = [x^n](x+x^2)^k/(1-x-x^2-x^3)^(k+1).

T(n,k) = sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j=0..n-k-2*i),i=0..n).

T(n,k) = sum(binomial(k,i)*(-1)^(k-i)*sum(binomial(j+k,k)*trinomial(i+j,n-3*k+2*i-j),j=0..n-k),i=0..k)

Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) + T(n+1,k) + T(n,k+1)

Example

Triangle begins:
1
1,1
2,3,1
4,7,5,1
7,17,16,7,1
13,38,46,29,9,1
24,82,122,99,46,11,1
44,174,304,303,184,67,13,1
81,362,728,857,641,309,92,15,1

Mathematica

Flatten[Table[Sum[Binomial[i+k, k]Sum[Binomial[i+k, j]Binomial[n-i-j, i+k], {j, 0, n-k-2i}], {i, 0, n}], {n, 0, 20}, {k, 0, n}]]

Prog

(Maxima) create_list(sum(binomial(i+k, k)*sum(binomial(i+k, j)*binomial(n-i-j, i+k), j, 0, n-k-2*i), i, 0, n), n, 0, 8, k, 0, n);

Crossrefs

Cf. A104580, A187889, A077936, A077946.

Sequence in context: A298364 A356769 A111776 * A299714 A171083 A364602

Adjacent sequences: A189184 A189185 A189186 * A189188 A189189 A189190

Keyword

nonn,easy,tabl

Author

Emanuele Munarini, Apr 18 2011

Extensions

a(23) and a(40) corrected by Georg Fischer, Feb 20 2021 and Apr 29 2022

Status

approved