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

1,2

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

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).

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

Example

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).

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 *)

Prog

(Magma) [Binomial(n+3, 4)-2*Binomial(n+2, 4)+ 3*Binomial(n+1, 4): n in [1..40]]; // Vincenzo Librandi, Feb 04 2013
(PARI) x='x+O('x^99); Vec(x*(1-2*x+3*x^2)/(1-x)^5) \\ Altug Alkan, Aug 16 2017

Crossrefs

Cf. A000332, A014628, A134392.

Sequence in context: A057765 A290866 A014628 * A034504 A191522 A140481

Adjacent sequences: A134390 A134391 A134392 * A134394 A134395 A134396

Keyword

nonn,easy

Author

Gary W. Adamson, Oct 23 2007

Status

approved