A036073 Triangle of coefficients arising in calculation of A002872 and A002874 (sorting numbers).

1, 2, 1, 5, 1, 6, 15, 1, 11, 30, 52, 1, 20, 80, 150, 203, 1, 37, 210, 525, 780, 877, 1, 70, 560, 1785, 3395, 4263, 4140, 1, 135, 1526, 6125, 14140, 22288, 24556, 21147, 1, 264, 4240, 21420, 58842, 109998, 150402, 149040, 115975, 1, 521, 11970, 76385, 248115

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf

Comments

For connection to A002872, A002874, and other columns of A162663, see the formula in A162663. - Andrey Zabolotskiy, Oct 25 2017

Examples

			Triangle begins:
  1;
  .  2;
  .  1,  5;
  .  1,  6,  15;
  .  1, 11,  30,  52;
  .  1, 20,  80, 150, 203;
  .  1, 37, 210, 525, 780, 877;
  ...
		

References

• T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

Links

• T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
• Index entries for sequences related to sorting

Crossrefs

Row sums give A001861.

Diagonal gives A000110(n+1) - Alois P. Heinz, Mar 27 2013

Cf. A162663.

Programs

Maple

egf:= exp(exp(x*y)+y*(exp(x)-1)-1):
T:= (n, k)-> n!*coeff(series(coeff(series(egf, y, k+1)
                , y, k), x, n+1), x, n):
seq(seq(T(n, k), k=min(n, 1)..n), n=0..10);  # Alois P. Heinz, Mar 28 2013

PARI

T(n, k) = { my(y = 'y + 'y*O('y^k), x = 'x + 'x*O('x^n); ); n!*polcoeff(polcoeff(exp(exp(x*y)+y*(exp(x)-1)-1), n, 'x), k, 'y); }
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* print triangle */
\\ Michel Marcus, Mar 27 2013

PARI

listpols(n)= {my(z = t + t*O(t^n)); zp = exp(exp(z)-1+(exp(p*z)-1)/p); for (i=0, n, print(i!*polcoeff(zp, i, t)););} \\ Michel Marcus, Mar 27 2013

Formula

E.g.f.: exp(exp(x*y)+y*(exp(x)-1)-1).

Extensions

Edited by Vladeta Jovovic, Sep 17 2003

Name corrected by Andrey Zabolotskiy, Oct 22 2017