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%I #39 Nov 18 2021 11:27:18
%S 1,1,1,1,3,2,2,9,13,6,5,32,72,69,24,16,132,409,605,432,120,61,623,
%T 2480,5016,5498,3120,720,271,3314,16222,41955,62626,54370,25560,5040,
%U 1372,19628,114594,363123,690935,814690,584580,234360,40320
%N Triangle read by rows: T(n,m) (n>=0, 1 <= m <= n+1) = number of unlabeled rigid interval posets with n non-maximal and m maximal elements.
%H Alois P. Heinz, <a href="/A193344/b193344.txt">Rows n = 0..50, flattened</a>
%H Soheir Mohamed Khamis, <a href="https://doi.org/10.1007/s11083-011-9213-5">Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height</a>, Order 29, pp. 443-461 (2012).
%F T(n,m) = [ y^n z^m ] W(y,z); W(y,z) = z + z*(W(y,y+z+yz) - W(y,z)).
%e Triangle begins
%e 1
%e 1 1
%e 1 3 2
%e 2 9 13 6
%e 5 32 72 69 24
%e 16 132 409 605 432 120
%e 61 623 2480 5016 5498 3120 720
%e 271 3314 16222 41955 62626 54370 25560 5040
%e 1372 19628 114594 363123 690935 814690 584580 ...
%p w:= proc(t) option remember;
%p `if`(t=0, 1, expand(convert(series(series(z+z*(subs(
%p z=z+y+y*z, w(t-1)) -w(t-1)), z, t+1), y, t+1), polynom)))
%p end:
%p T:= (n,m)-> coeff(coeff(w(m+n), z, m), y, n):
%p seq(seq(T(n, m), m=1..n+1), n=0..10); # _Alois P. Heinz_, Aug 27 2011
%t w[t_] := w[t] = If[t == 0, 1, Expand[Normal[Series[Series[z+z*((w[t-1] /. z -> z+y+y*z)-w[t-1]), {z, 0, t+1}], {y, 0, t+1}]]]]; T[n_, m_] := Coefficient[Coefficient[w[m+n], z, m], y, n]; Table[Table[T[n, m], {m, 1, n+1}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 05 2014, after _Alois P. Heinz_ *)
%Y First column is A138265, second column is A194530.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Aug 26 2011
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