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A081579 Pascal-(1,4,1) array. 11
1, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 16, 46, 16, 1, 1, 21, 106, 106, 21, 1, 1, 26, 191, 396, 191, 26, 1, 1, 31, 301, 1011, 1011, 301, 31, 1, 1, 36, 436, 2076, 3606, 2076, 436, 36, 1, 1, 41, 596, 3716, 9726, 9726, 3716, 596, 41, 1, 1, 46, 781, 6056, 21746, 33876, 21746, 6056 (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 A016861, A081587, A081588. Coefficients of the row polynomials in the Newton basis are given by A013612.

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)+4T(n-1, k-1)+T(n-1, k). Rows are the expansions of (1+4x)^k/(1-x)^(k+1).

As a number triangle, this is the Riordan array (1/(1-x), x*(1+4*x)/(1-x)). It has row sums A063727(n). - Philippe Deléham, Mar 15 2014

E.g.f. for the n-th subdiagonal of the triangle, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(5*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 10*x + 25*x^2/2) = 1 + 11*x + 46*x^2/2! + 106*x^3/3! + 191*x^4/4! + 301*x^5/5! + .... - Peter Bala, Mar 05 2017

EXAMPLE

Rows start

1 1 1 1 1 ...

1 6 11 16 21 ...

1 11 46 106 191 ...

1 16 106 396 1011 ...

1 21 191 1011 3606 ...

As triangle this begins:

1;

1, 1;

1, 6, 1;

1, 11, 11, 1;

1, 16, 46, 16, 1;

1, 21, 106, 106, 21, 1;

1, 26, 191, 396, 191, 26, 1;

1, 31, 301, 1011, 1011, 301, 31, 1;

1, 36, 436, 2076, 3606, 2076, 436, 36, 1;

1, 41, 596, 3716, 9726, 9726, 3716, 596, 41, 1;

1, 46, 781, 6056, 21746, 33876, 21746, 6056, 781, 46, 1; - Philippe Deléham, Mar 15 2014

MATHEMATICA

Table[ Hypergeometric2F1[-k, k-n, 1, 5], {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), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Sequence in context: A046621 A046617 A131063 * A295707 A146772 A202868

Adjacent sequences:  A081576 A081577 A081578 * A081580 A081581 A081582

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Mar 23 2003

STATUS

approved

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Last modified July 9 14:56 EDT 2020. Contains 335543 sequences. (Running on oeis4.)