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

%I A036073 #44 Aug 14 2021 11:06:40
%S A036073 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 A036073 560,1785,3395,4263,4140,1,135,1526,6125,14140,22288,24556,21147,1,
%U A036073 264,4240,21420,58842,109998,150402,149040,115975,1,521,11970,76385,248115
%N A036073 Triangle of coefficients arising in calculation of A002872 and A002874 (sorting numbers).
%C A036073 For connection to A002872, A002874, and other columns of A162663, see the formula in A162663. - _Andrey Zabolotskiy_, Oct 25 2017
%D A036073 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%H A036073 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 A036073 <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F A036073 E.g.f.: exp(exp(x*y)+y*(exp(x)-1)-1).
%e A036073 Triangle begins:
%e A036073   1;
%e A036073   .  2;
%e A036073   .  1,  5;
%e A036073   .  1,  6,  15;
%e A036073   .  1, 11,  30,  52;
%e A036073   .  1, 20,  80, 150, 203;
%e A036073   .  1, 37, 210, 525, 780, 877;
%e A036073   ...
%p A036073 egf:= exp(exp(x*y)+y*(exp(x)-1)-1):
%p A036073 T:= (n, k)-> n!*coeff(series(coeff(series(egf, y, k+1)
%p A036073                 , y, k), x, n+1), x, n):
%p A036073 seq(seq(T(n, k), k=min(n, 1)..n), n=0..10);  # _Alois P. Heinz_, Mar 28 2013
%o A036073 (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 A036073 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* print triangle */
%o A036073 \\ _Michel Marcus_, Mar 27 2013
%o A036073 (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 A036073 Row sums give A001861.
%Y A036073 Diagonal gives A000110(n+1) - _Alois P. Heinz_, Mar 27 2013
%Y A036073 Cf. A162663.
%K A036073 nonn,tabf
%O A036073 0,2
%A A036073 _N. J. A. Sloane_
%E A036073 Edited by _Vladeta Jovovic_, Sep 17 2003
%E A036073 Name corrected by _Andrey Zabolotskiy_, Oct 22 2017