A113582 Triangle T(n,m) read by rows: T(n,m) = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 10, 7, 1, 1, 11, 19, 19, 11, 1, 1, 16, 31, 37, 31, 16, 1, 1, 22, 46, 61, 61, 46, 22, 1, 1, 29, 64, 91, 101, 91, 64, 29, 1, 1, 37, 85, 127, 151, 151, 127, 85, 37, 1, 1, 46, 109, 169, 211, 226, 211, 169, 109, 46, 1

Offset: 1

Author

Roger L. Bagula, Aug 25 2008

Keywords

• nonn
• tabl
• easy

Comments

From Paul Barry, Jan 07 2009: (Start)

This triangle follows a general construction method as follows: Let a(n) be an integer sequence with a(0)=1, a(1)=1. Then T(n,k,r) := [k<=n](1+r*a(k)*a(n-k)) defines a symmetrical triangle.

Row sums are n + 1 + r*Sum_{k=0..n} a(k)*a(n-k) and central coefficients are 1+r*a(n)^2.

Here a(n) = C(n+1,2) and r=1.

Row sums are A154322 and central coefficients are A154323. (End)

Examples

			{1},
{1, 1},
{1, 2, 1},
{1, 4, 4, 1},
{1, 7, 10, 7, 1},
{1, 11, 19, 19, 11, 1},
{1, 16, 31, 37, 31, 16, 1},
{1, 22, 46, 61, 61, 46, 22, 1},
{1, 29, 64, 91, 101, 91, 64, 29, 1},
{1, 37, 85, 127, 151, 151, 127, 85, 37, 1},
{1, 46, 109, 169, 211, 226, 211, 169, 109, 46, 1}
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Programs

Magma

/* As triangle: */ [[(n-m)*(n-m+1)*m*(m+1)/4+1: m in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 12 2016

Mathematica

t[n_, m_] = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]//Flatten

PARI

for(n=0,15, for(k=0,n, print1((n-k)*(n-k+1)*k*(k+1)/4 + 1, ", "))) \\ G. C. Greubel, Aug 31 2018

Formula

T(n,m) = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1.