A131820 - OEIS

%I #30 Apr 11 2023 08:24:16

%S 1,6,16,33,59,96,146,211,293,394,516,661,831,1028,1254,1511,1801,2126,

%T 2488,2889,3331,3816,4346,4923,5549,6226,6956,7741,8583,9484,10446,

%U 11471,12561,13718,14944,16241,17611,19056,20578,22179,23861,25626

%N Row sums of triangle A131819.

%C Let M(n) be the n-th square matrix whose (i,j)-entry equals i^2/(i^2+1) if i=j and equals 1 otherwise. Then a(n) = (-1)^(n+1) * gamma(1-i+n) * gamma(1+i+n) * sinh(Pi)/Pi times the determinant of M(n). - _John M. Campbell_, Sep 07 2011

%F Binomial transform of (1, 5, 5, 2, 0, 0, 0, ...).

%F From _Alois P. Heinz_, May 04 2009: (Start)

%F a(n) = n^3/3 + n^2/2 + (7/6)*n - 1.

%F a(n) = -1 + Sum_{k=1..n} (k^2+1).

%F a(n) = A000330(n) + A000027(n) - A000012(n).

%F G.f.: (2*x^3 - 4*x^2 + 5*x - 1) / (x-1)^4. (End)

%F a(n) = n^2 + a(n-1) + 1, n > 1. - _Gary Detlefs_, Jun 29 2010

%F From _Gary Detlefs_, Jun 30 2010: (Start)

%F a(n) = (2n^3 + 3n^2 + 7n - 6)/6, n > 0.

%F a(n) = A081489(n) + A005563(n-1), n > 0. (End)

%e a(4) = 33 = (1, 3, 3, 1) dot (1, 5, 5, 2) = (1 + 15 + 15 + 2).

%e a(4) = 33 = sum of row 4 terms of triangle A131819: (13 + 9 + 7 + 4).

%p a:= n-> (7+(3+2*n)*n)*n/6-1:

%p seq(a(n), n=1..40);  # _Alois P. Heinz_, May 04 2009

%t Table[n^3/3 + n^2/2 + 7*n/6 - 1, {n, 100}]

%Y Cf. A131819, A000330, A081489, A005563.

%K nonn

%O 1,2

%A _Gary W. Adamson_, Jul 18 2007

%E More terms from _Alois P. Heinz_, May 04 2009