OFFSET
0,5
COMMENTS
LINKS
Vincenzo Librandi, Rows n = 0..100, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
FORMULA
Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 5*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+5*x)^k/(1-x)^(k+1).
From Paul Barry, Aug 28 2008: (Start)
Number triangle T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*5^j.
Riordan array (1/(1-x), x*(1+5*x)/(1-x)). (End)
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 6). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(6*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 12*x + 36*x^2/2) = 1 + 13*x + 61*x^2/2! + 145*x^3/3! + 265*x^4/4! + 421*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n, k, 3) = A002532(n+1). - G. C. Greubel, May 26 2021
EXAMPLE
Square array begins as:
1, 1, 1, 1, 1, ... A000012;
1, 7, 13, 19, 25, ... A016921;
1, 13, 61, 145, 265, ... A081589;
1, 19, 145, 595, 1585, ... A081590;
1, 25, 265, 1585, 6145, ...
The triangle begins as:
1;
1, 1;
1, 7, 1;
1, 13, 13, 1;
1, 19, 61, 19, 1;
1, 25, 145, 145, 25, 1;
1, 31, 265, 595, 265, 31, 1;
1, 37, 421, 1585, 1585, 421, 37, 1;
1, 43, 613, 3331, 6145, 3331, 613, 43, 1;
1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1;
1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1; - Philippe Deléham, Mar 15 2014
MATHEMATICA
Table[Hypergeometric2F1[-k, k-n, 1, 6], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, May 24 2013 *)
PROG
(Magma)
A081580:= func< n, k, q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081580(n, k, 5): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021
(Sage) flatten([[hypergeometric([-k, k-n], [1], 6).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 23 2003
STATUS
approved