A134393 Row sums of triangle A134392.

1, 3, 8, 20, 45, 91, 168, 288, 465, 715, 1056, 1508, 2093, 2835, 3760, 4896, 6273, 7923, 9880, 12180, 14861, 17963, 21528, 25600, 30225, 35451, 41328, 47908, 55245, 63395, 72416, 82368, 93313, 105315, 118440, 132756, 148333, 165243, 183560, 203360, 224721, 247723, 272448

Offset: 1

Author

Gary W. Adamson, Oct 23 2007

Keywords

• nonn
• easy

Comments

Binomial transform of [1, 2, 3, 4, 2, 0, 0, 0, ...].

The Kn4 triangle sums of A139600 are given by this sequence. For the definitions of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

Examples

			a(4) = 20 = (1, 3, 3, 1) dot (1, 2, 3, 4) = (1 + 6 + 9 + 4).
a(4) = sum of row 4 terms of triangle A134392: (8 + 7 + 4 + 1).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000332, A014628, A134392.

Programs

Magma

[Binomial(n+3, 4)-2*Binomial(n+2, 4)+ 3*Binomial(n+1, 4): n in [1..40]]; // Vincenzo Librandi, Feb 04 2013

Mathematica

Table[(n^4 - 2*n^3 + 5*n^2 + 8*n)/12, {n, 1, 40}] (* Vincenzo Librandi, Feb 04 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,3,8,20,45},50] (* Harvey P. Dale, May 28 2018 *)

PARI

x='x+O('x^99); Vec(x*(1-2*x+3*x^2)/(1-x)^5) \\ Altug Alkan, Aug 16 2017

Formula

From R. J. Mathar, Jun 08 2008: (Start)

O.g.f.: x*(1-2*x+3*x^2)/(1-x)^5.

a(n) = A014628(n+1). (End)

a(n) = binomial(n+3,4) - 2*binomial(n+2,4) + 3*binomial(n+1,4). - Johannes W. Meijer, Apr 29 2011, corrected by Eric Rowland, Aug 16 2017

a(n) = n*(n + 1)*(n^2 - 3*n + 8)/12. - Johannes W. Meijer, Apr 29 2011, corrected by Eric Rowland, Aug 16 2017