A081579 Pascal-(1,4,1) array.

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

Offset: 0

Author

Paul Barry, Mar 23 2003

Keywords

• easy
• nonn
• tabl

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.

Examples

			Square array begins as:
  1,  1,   1,    1,    1, ... A000012;
  1,  6,  11,   16,   21, ... A016861;
  1, 11,  46,  106,  191, ... A081587;
  1, 16, 106,  396, 1011, ... A081588;
  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
		

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.

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).

Cf. A016861, A063727, A081587, A081588.

Programs

Magma

A081579:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081579(n,k,4): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Mathematica

Table[Hypergeometric2F1[-k, k-n, 1, 5], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

Sage

flatten([[hypergeometric([-k, k-n], [1], 5).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Formula

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 4*T(n-1, k-1) + T(n-1, k).

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

From Philippe Deléham, Mar 15 2014: (Start)

Riordan array (1/(1-x), x*(1+4*x)/(1-x)).

Sum_{k=0..n} T(n, k) = A063727(n). (End)

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

From G. C. Greubel, May 26 2021: (Start)

T(n, k, m) = Hypergeometric2F1([-k, k-n], [1], m+1), for m = 4.

T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 4. (End)