A239098 Triangle read by rows: T(0,0)=1; T(m,0)=0; otherwise T(m,n) = (m-1)*T(m-1,n)+(m-1+n)*T(m-1,n-1).

1, 0, 1, 0, 1, 3, 0, 2, 10, 15, 0, 6, 40, 105, 105, 0, 24, 196, 700, 1260, 945, 0, 120, 1148, 5068, 12600, 17325, 10395, 0, 720, 7848, 40740, 126280, 242550, 270270, 135135, 0, 5040, 61416, 363660, 1332100, 3213210, 5045040, 4729725, 2027025, 0, 40320, 541728, 3584856, 15020720, 43022980, 85345260, 113513400, 91891800, 34459425

Offset: 0

Author

N. J. A. Sloane, Mar 23 2014

Keywords

• nonn
• tabl

Comments

If the first column is omitted we get A075856, which has much more information about this triangle.

Examples

			Triangle begins:
1,
0, 1,
0, 1, 3,
0, 2, 10, 15,
0, 6, 40, 105, 105,
0, 24, 196, 700, 1260, 945,
0, 120, 1148, 5068, 12600, 17325, 10395,
0, 720, 7848, 40740, 126280, 242550, 270270, 135135,
...
		

References

• P. W. Shor, Problem 78-6: A combinatorial identity, in Problems and Solutions column, SIAM Review; problem in 20, p. 394 (1978); solution in 21, pp. 258-260 (1979).

Links

• Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.

Crossrefs

Cf. A075856.

Programs

Maple

T:=proc(m,n) option remember;
if (m=0) and (n=0) then 1;
elif (m=0) or (n=0) then 0;
else (m-1)*T(m-1,n)+(m-1+n)*T(m-1,n-1); fi; end;
M:=20;
for m from 0 to M do
lprint([seq(T(m,n),n=0..m)]); od: