A131819 -id:A131819 - cp's OEIS frontend

1, 6, 16, 33, 59, 96, 146, 211, 293, 394, 516, 661, 831, 1028, 1254, 1511, 1801, 2126, 2488, 2889, 3331, 3816, 4346, 4923, 5549, 6226, 6956, 7741, 8583, 9484, 10446, 11471, 12561, 13718, 14944, 16241, 17611, 19056, 20578, 22179, 23861, 25626

Offset: 1

Gary W. Adamson, Jul 18 2007

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

			a(4) = 33 = (1, 3, 3, 1) dot (1, 5, 5, 2) = (1 + 15 + 15 + 2).
a(4) = 33 = sum of row 4 terms of triangle A131819: (13 + 9 + 7 + 4).

Cf. A131819, A000330, A081489, A005563.

a:= n-> (7+(3+2*n)*n)*n/6-1:
seq(a(n), n=1..40);  # Alois P. Heinz, May 04 2009

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

Binomial transform of (1, 5, 5, 2, 0, 0, 0, ...).
From Alois P. Heinz, May 04 2009: (Start)
a(n) = n^3/3 + n^2/2 + (7/6)*n - 1.
a(n) = -1 + Sum_{k=1..n} (k^2+1).
a(n) = A000330(n) + A000027(n) - A000012(n).
G.f.: x*(1 + 2*x - 2*x^2 + x^3)/(1 - x)^4. (End)
a(n) = n^2 + a(n-1) + 1, n > 1. - Gary Detlefs, Jun 29 2010
From Gary Detlefs, Jun 30 2010: (Start)
a(n) = (2n^3 + 3n^2 + 7n - 6)/6, n > 0.
a(n) = A081489(n) + A005563(n-1), n > 0. (End)
E.g.f.: 1 + exp(x)*(2*x^3 + 9*x^2 + 12*x - 6)/6. - Stefano Spezia, Mar 02 2025

More terms from Alois P. Heinz, May 04 2009

cp's OEIS Frontend

A131820 Row sums of triangle A131819.

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