A129858 A triangle of coefficients based on A000217: a(n)=Binomial[n+2,2]; t(n,m)=a(n - m + 1)*a(m + 1) - a((n - m + 1)*(m + 1)).

6, 12, 12, 20, 21, 20, 30, 32, 32, 30, 42, 45, 45, 45, 42, 56, 60, 59, 59, 60, 56, 72, 77, 74, 72, 74, 77, 72, 90, 96, 90, 84, 84, 90, 96, 90, 110, 117, 107, 95, 90, 95, 107, 117, 110, 132, 140, 125, 105, 92, 92, 105, 125, 140, 132, 156, 165, 144, 114, 90, 81, 90, 114, 144

Offset

1,1

Comments

Row sums are:

{6, 24, 61, 124, 219, 350, 518, 720, 948, 1188, 1419}.

References

G. E. Andrews, Number Theory, 1971, Dover Publications New York, p 44,p 85.

Links

Table of n, a(n) for n=1..64.

Formula

a(n)=Binomial[n+2,2]; t(n,m)=a(n - m + 1)*a(m + 1) - a((n - m + 1)*(m + 1)).

Example

{6},
{12, 12},
{20, 21, 20},
{30, 32, 32, 30},
{42, 45, 45, 45, 42},
{56, 60, 59, 59, 60, 56},
{72, 77, 74, 72, 74, 77, 72},
{90, 96, 90, 84, 84, 90, 96, 90},
{110, 117, 107, 95, 90, 95, 107, 117, 110},
{132, 140, 125, 105, 92, 92, 105, 125, 140, 132},
{156, 165, 144, 114, 90, 81, 90, 114, 144, 165, 156}

Mathematica

Clear[a, n, m, t] (*A000217*) a[0] = 1; a[1] = 3; a[n_] := a[n] = Binomial[n + 2, 2]; Table[a[n], {n, 0, 30}]; t[n_, m_] = FullSimplify[a[n - m + 1]*a[m + 1] - a[(n - m + 1)*(m + 1)]]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Crossrefs

Cf. A000217.

Sequence in context: A315581 A315582 A315583 * A061928 A315584 A315585

Adjacent sequences: A129855 A129856 A129857 * A129859 A129860 A129861

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Aug 25 2008

Status

approved