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A134393
Row sums of triangle A134392.
4
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
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
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
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Oct 23 2007
STATUS
approved

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Last modified September 20 06:06 EDT 2024. Contains 376027 sequences. (Running on oeis4.)