A122890 Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields the g.f. of column n in the triangle A122888, for n>=1.

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 10, 14, 0, 0, 0, 8, 70, 42, 0, 0, 0, 4, 160, 424, 132, 0, 0, 0, 1, 250, 1978, 2382, 429, 0, 0, 0, 0, 302, 6276, 19508, 12804, 1430, 0, 0, 0, 0, 298, 15674, 106492, 168608, 66946, 4862, 0, 0, 0, 0, 244, 33148, 451948, 1445208, 1337684, 343772, 16796

Offset: 0

Author

Paul D. Hanna, Sep 18 2006

Keywords

• nonn
• tabl

Comments

Main diagonal forms the Catalan numbers (A000108). Row sums gives the factorials. In table A122888, row n lists the coefficients of x^k, k = 1..2^n, in the n-th self-composition of (x + x^2) for n >= 0.

Parker gave the following combinatorial interpretation of the numbers: For n > 0, T(n, j) is the number of sequences c_1c_2...c_n of positive integers such that 1 <= c_i <= i for each i in {1, 2, .., n} with exactly j - 1 values of i such that c_i <= c_{i+1}. - Peter Luschny, May 05 2013

Examples

			Triangle begins:
1;
0,1;
0,0,2;
0,0,1,5;
0,0,0,10,14;
0,0,0,8,70,42;
0,0,0,4,160,424,132;
0,0,0,1,250,1978,2382,429;
0,0,0,0,302,6276,19508,12804,1430;
0,0,0,0,298,15674,106492,168608,66946,4862;
0,0,0,0,244,33148,451948,1445208,1337684,343772,16796;
0,0,0,0,162,61806,1614906,9459090,16974314,10003422,1744314,58786;
0,0,0,0,84,103932,5090124,51436848,161380816,180308420,71692452,8780912,208012; ...
Table A122888 starts:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
where row n gives the g.f. of the n-th self-composition of (x+x^2).
From _Paul D. Hanna_, Apr 11 2009: (Start)
ROW-REVERSAL yields triangle A158830:
1;
1, 0;
2, 0, 0;
5, 1, 0, 0;
14, 10, 0, 0, 0;
42, 70, 8, 0, 0, 0;
132, 424, 160, 4, 0, 0, 0;
429, 2382, 1978, 250, 1, 0, 0, 0; ...
where
g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825];
g.f. of row n of A158825 = n-th iteration of x*Catalan(x).
RELATED ARRAY A158825 begins:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,...;
1,4,20,110,640,3870,24084,153306,993978,...;
1,5,30,195,1330,9380,67844,500619,3755156,...;
1,6,42,315,2464,19852,163576,1372196,11682348,...;
1,7,56,476,4200,38052,351792,3305484,31478628,...;
1,8,72,684,6720,67620,693048,7209036,75915708,...; ...
which consists of successive iterations of x*Catalan(x).
(End)
		

Links

• Toufik Mansour, Mark Shattuck, Statistics on bargraphs of inversion sequences of permutations, Discrete Math. Lett. (2020) Vol. 4, 42-49.
• Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times, arXiv:1704.04288 [math.CO], 2017.
• Susan Field Parker, The Combinatorics of Functional Composition and Inversion, Ph.D. Dissertation, Brandeis Univ. (1993) (Section 2.3.4, p. 27,28.)

Crossrefs

Cf. A122888; A122891 (column sums); diagonals: A122892, A000108.

Cf. related tables: A158830, A158825. [Paul D. Hanna, Apr 11 2009]

Programs

Mathematica

nmax = 11;
f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand; T = Table[ SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];
row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x];
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Jul 13 2018 *)

Formula

From Paul D. Hanna, Apr 11 2009: (Start)

G.f. of row n: (1-x)^n*[g.f. of column n of A122888] where the g.f. of row n of A122888 is the n-th iteration of x+x^2.

Row-reversal forms triangle A158830 where g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825], and the g.f. of row n of array A158825 is the n-th iteration of x*C(x) and C(x) is the g.f. of the Catalan sequence A000108. (End)