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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

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.

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)

EXAMPLE

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], ...

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 *)

PROG

(PARI) {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)))}

(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 */

(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

(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

CROSSREFS

Cf. A077939, A103141.

Sequence in context: A124802 A211350 A178867 * A121524 A103141 A129818

Adjacent sequences:  A102033 A102034 A102035 * A102037 A102038 A102039

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 30 2004

STATUS

approved

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Last modified January 22 12:13 EST 2019. Contains 319363 sequences. (Running on oeis4.)