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A081580 Pascal-(1,5,1) array. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016921, A081589, A081590. Coefficients of the row polynomials in the Newton basis are given by A013613.

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)+5T(n-1, k-1)+T(n-1, k). Rows are the expansions of (1+5x)^k/(1-x)^(k+1).

Number triangle T(n,k) = sum{j=0..n-k, binomial(n-k,j)binomial(k,j)6^j}; Riordan array (1/(1-x),x(1+5x)/(1-x)); [Paul Barry, Aug 28 2008]

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

EXAMPLE

Rows start

1 1 1 1 1 ...

1 7 13 19 25 ...

1 13 61 145 265 ...

1 19 145 595 1585 ...

1 25 265 1585 6145 ...

As a triangle this begins:

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; etc.

- 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 *)

CROSSREFS

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Sequence in context: A273506 A287326 A131065 * A082110 A275526 A141597

Adjacent sequences:  A081577 A081578 A081579 * A081581 A081582 A081583

KEYWORD

nonn,tabl,easy

AUTHOR

Paul Barry, Mar 23 2003

STATUS

approved

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Last modified February 18 03:33 EST 2020. Contains 332006 sequences. (Running on oeis4.)