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A103141 Riordan array (1/(1-x), x(1+x+x^2+x^3)/(1-x)). 2
1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 15, 7, 1, 1, 14, 35, 28, 9, 1, 1, 18, 68, 84, 45, 11, 1, 1, 22, 116, 207, 165, 66, 13, 1, 1, 26, 180, 441, 491, 286, 91, 15, 1, 1, 30, 260, 840, 1251, 996, 455, 120, 17, 1, 1, 34, 356, 1464, 2823, 2948, 1814, 680, 153, 19, 1, 1, 38, 468, 2376 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Generalized Pascal matrix : row sums are generalized Pell numbers A103142 and diagonal sums are the Pentanacci numbers A001591(n+4). One of a family of generalized Pascal triangles given by the Riordan arrays (1/(1-x), x*Sum_{j=0..k} x^k/(1-x)). This array has the 'k+2-nacci' numbers as diagonal sums and generalized Pell numbers b(n) = 2b(n-1) + Sum_{j=1..k} b(n-1-j) as row sums. The first two arrays of the family are Pascal's triangle and the Delannoy number triangle.

LINKS

Table of n, a(n) for n=0..69.

FORMULA

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) + T(n-4, k-1), with T(0, 0)=1.

G.f.: 1/(1-y-x*(1+y+y^2+y^3). - Vladimir Kruchinin, Apr 21 2015

EXAMPLE

Triangle begins

1,

1,  1,

1,  3,   1,

1,  6,   5,  1,

1, 10,  15,  7,   1,

1, 14,  35,  28,  9, 1,

1, 18,  68,  84,  45, 11, 1,

1, 22, 116, 207, 165, 66, 13, 1,

1, 26, 180, 441, 491, 286, 91, 15, 1,

1, 30, 260, 840, 1251, 996, 455, 120, 17, 1,

1, 34, 356, 1464, 2823, 2948, 1814, 680, 153, 19, 1, ...

MATHEMATICA

T[_?Positive, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; 0<k<n := T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k-1] + T[n-3, k-1] + T[n-4, k-1]; T[_, _] = 0; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 24 2017 *)

PROG

(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. A102036.

Sequence in context: A178867 A102036 A121524 * A129818 A085478 A123970

Adjacent sequences:  A103138 A103139 A103140 * A103142 A103143 A103144

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jan 24 2005

STATUS

approved

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Last modified February 17 22:32 EST 2018. Contains 299297 sequences. (Running on oeis4.)