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A036073 Triangle of coefficients arising in calculation of A002872 and A002874 (sorting numbers). 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 2 17:41 EST 2021. Contains 349445 sequences. (Running on oeis4.)