A102036 Triangle, read by rows, where the terms are generated by the rule: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-3,k-1), with T(0,0)=1.

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 9, 15, 7, 1, 1, 12, 33, 28, 9, 1, 1, 15, 60, 81, 45, 11, 1, 1, 18, 96, 189, 161, 66, 13, 1, 1, 21, 141, 378, 459, 281, 91, 15, 1, 1, 24, 195, 675, 1107, 946, 449, 120, 17, 1, 1, 27, 258, 1107, 2349, 2673, 1742, 673, 153, 19, 1

Offset: 0

Author

Paul D. Hanna, Dec 30 2004

Keywords

• nonn
• tabl

Comments

Row sums form A077939. This sequence was inspired by Luke Hanna.

Diagonal sums are A000078(n+3). - Philippe Deléham, Feb 16 2014

Riordan array (1/(1-x), x*(1+x+x^2)/(1-x)). - Philippe Deléham, Feb 16 2014

Examples

			Generated by adding preceding terms in the triangle at positions that form the letter 'L':
T(n,k) =
T(n-3,k-1) +
T(n-2,k-1) +
T(n-1,k-1) + T(n-1,k).
Rows begin:
  [1],
  [1,  1],
  [1,  3,   1],
  [1,  6,   5,   1],
  [1,  9,  15,   7,   1],
  [1, 12,  33,  28,   9,   1],
  [1, 15,  60,  81,  45,  11,  1],
  [1, 18,  96, 189, 161,  66, 13,  1],
  [1, 21, 141, 378, 459, 281, 91, 15, 1], ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Kuhapatanakul, Kantaphon; Anantakitpaisal, Pornpawee The k-nacci triangle and applications. Cogent Math. 4, Article ID 1333293, 13 p. (2017).
• J. L. Ramírez, V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.

Crossrefs

Cf. A077939, A103141.

Programs

Magma

[[(&+[Binomial(n-m,k)*(&+[Binomial(j,m-j)*Binomial(k,j):j in [0..k]]): m in [0..n-k]]): k in [0..n]]: n in [0..15]]; // G. C. Greubel, Dec 11 2018

Maple

T:=(n,k)->add(add((binomial(j,m-j)*binomial(k,j))*binomial(n-m,k),j=0..k),m=0..n-k): seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Dec 11 2018

Mathematica

T[n_, k_] := If[n < k || k < 0, 0, If[n == 0, 1, T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k - 1] + T[n - 3, k - 1]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 07 2018 *)
Table[Sum[Binomial[n-m, k]*Sum[Binomial[j, m-j]*Binomial[k, j], {j, 0, k}], {m, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 11 2018 *)

Maxima

T(n,k):=sum((sum(binomial(j,m-j)*binomial(k,j),j,0,k))*binomial(n-m,k),m,0,n-k); /* Vladimir Kruchinin, Apr 21 2015 */

PARI

{T(n,k)=if(n
				

Sage

[[sum(binomial(n-m,k)*sum(binomial(j,m-j)*binomial(k,j) for j in (0..k)) for m in (0..n-k)) for k in (0..n)] for n in range(15)] # G. C. Greubel, Dec 11 2018

Formula

G.f.: 1/(1-y-x*(1+y+y^2)). - Vladimir Kruchinin, Apr 21 2015

T(n,k) = Sum_{m=0..(n-k)} (Sum_{j=0..k} C(j,m-j)*C(k,j))*C(n-m,k). - Vladimir Kruchinin, Apr 21 2015

From Werner Schulte, Dec 07 2018: (Start)

G.f. of column k: Sum_{n>=0} T(n+k,k) * x^n = (1+x+x^2)^k / (1-x)^(k+1) = (1-x^3)^k / (1-x)^(2*k+1).

Let k >= 0 be some fixed integer and a_k(n) be multiplicative with a_k(p^e) = T(e+k,k) for prime p and e >= 0. Then we have the Dirichlet g.f.: Sum{n>0} a_k(n) / n^s = (zeta(s))^(2*k+1) / (zeta(3*s))^k. (End)