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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. 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 (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

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

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

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],...

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 */

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 February 21 04:22 EST 2018. Contains 299389 sequences. (Running on oeis4.)