A036073 - OEIS

%I #44 Aug 14 2021 11:06:40

%S 1,2,1,5,1,6,15,1,11,30,52,1,20,80,150,203,1,37,210,525,780,877,1,70,

%T 560,1785,3395,4263,4140,1,135,1526,6125,14140,22288,24556,21147,1,

%U 264,4240,21420,58842,109998,150402,149040,115975,1,521,11970,76385,248115

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

%C For connection to A002872, A002874, and other columns of A162663, see the formula in A162663. - _Andrey Zabolotskiy_, Oct 25 2017

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

%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]

%H <a href="/index/So#sorting">Index entries for sequences related to sorting</a>

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

%e Triangle begins:

%e   1;

%e   .  2;

%e   .  1,  5;

%e   .  1,  6,  15;

%e   .  1, 11,  30,  52;

%e   .  1, 20,  80, 150, 203;

%e   .  1, 37, 210, 525, 780, 877;

%e   ...

%p egf:= exp(exp(x*y)+y*(exp(x)-1)-1):

%p T:= (n, k)-> n!*coeff(series(coeff(series(egf, y, k+1)

%p                 , y, k), x, n+1), x, n):

%p seq(seq(T(n, k), k=min(n, 1)..n), n=0..10);  # _Alois P. Heinz_, Mar 28 2013

%o (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); }

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* print triangle */

%o \\ _Michel Marcus_, Mar 27 2013

%o (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

%Y Row sums give A001861.

%Y Diagonal gives A000110(n+1) - _Alois P. Heinz_, Mar 27 2013

%Y Cf. A162663.

%K nonn,tabf

%O 0,2

%A _N. J. A. Sloane_

%E Edited by _Vladeta Jovovic_, Sep 17 2003

%E Name corrected by _Andrey Zabolotskiy_, Oct 22 2017