A174640 A triangular sequence:t(n,m)=A033306(n,m)-A033306(n,0)+1.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 10, 6, 1, 1, 24, 49, 49, 24, 1, 1, 110, 248, 298, 248, 110, 1, 1, 545, 1308, 1749, 1749, 1308, 545, 1, 1, 2877, 7229, 10421, 11611, 10421, 7229, 2877, 1, 1, 16114, 41998, 64114, 77134, 77134, 64114, 41998, 16114, 1, 1, 95496

Offset: 0

Author

Roger L. Bagula, Mar 25 2010

Keywords

• nonn
• tabl
• uned

Comments

Row sums are:

1, 2, 3, 6, 24, 148, 1016, 7206, 52667, 398722, 3137084,...

Examples

			{1},
{1, 1},
{1, 1, 1},
{1, 2, 2, 1},
{1, 6, 10, 6, 1},
{1, 24, 49, 49, 24, 1},
{1, 110, 248, 298, 248, 110, 1},
{1, 545, 1308, 1749, 1749, 1308, 545, 1},
{1, 2877, 7229, 10421, 11611, 10421, 7229, 2877, 1},
{1, 16114, 41998, 64114, 77134, 77134, 64114, 41998, 16114, 1},
{1, 95496, 256626, 410226, 523476, 565434, 523476, 410226, 256626, 95496, 1}
		

References

• This notebook downloaded from https://mathworld.wolfram.com/notebooks/Combinatorics/BellNumber.nb.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 80.

Crossrefs

A033306

Programs

Mathematica

b[0] := 1;
b[n_] := b[n] = Total[Table[b[k]Binomial[n - 1, k], {k, 0, n - 1}]];
a = b /@ Range[0, 70];
t[n_, m_] := Binomial[n, m]*a[[m + 1]]*a[[n - m + 1]];
Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,m)=A033306(n,m)-A033306(n,0)+1