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

0,2

Comments

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

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

Table of n, a(n) for n=0..50.

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

Formula

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

Example

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

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

Prog

(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

Crossrefs

Row sums give A001861.

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

Cf. A162663.

Sequence in context: A014648 A260147 A263454 * A124227 A064865 A178472

Adjacent sequences: A036070 A036071 A036072 * A036074 A036075 A036076

Keyword

nonn,tabf

Author

N. J. A. Sloane

Extensions

Edited by Vladeta Jovovic, Sep 17 2003

Name corrected by Andrey Zabolotskiy, Oct 22 2017

Status

approved