A158830 - OEIS

%I #17 Oct 26 2018 11:47:13

%S 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,

%T 429,2382,1978,250,1,0,0,0,1430,12804,19508,6276,302,0,0,0,0,4862,

%U 66946,168608,106492,15674,298,0,0,0,0,16796,343772,1337684,1445208,451948

%N Triangle, read by rows n>=1, where row n is the n-th differences of column n of array A158825, where the g.f. of row n of A158825 is the n-th iteration of x*Catalan(x).

%H Paul D. Hanna, <a href="/A158830/b158830.txt">Table of n, a(n), n = 1..1326 (rows 1..51).</a>

%H Toufik Mansour, Howard Skogman, Rebecca Smith, <a href="https://arxiv.org/abs/1704.04288">Passing through a stack k times</a>, arXiv:1704.04288 [math.CO], 2017.

%F Row sums equal the factorial numbers.

%F G.f. of row n = (1-x)^n*[g.f. of column n of A158825] where 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.

%F Row-reversal is triangle A122890 where g.f. of row n of A122890 = (1-x)^n*[g.f. of column n of A122888], and the g.f. of row n of array A122888 is the n-th iteration of x+x^2.

%e Triangle begins:

%e .1;

%e .1,0;

%e .2,0,0;

%e .5,1,0,0;

%e .14,10,0,0,0;

%e .42,70,8,0,0,0;

%e .132,424,160,4,0,0,0;

%e .429,2382,1978,250,1,0,0,0;

%e .1430,12804,19508,6276,302,0,0,0,0;

%e .4862,66946,168608,106492,15674,298,0,0,0,0;

%e .16796,343772,1337684,1445208,451948,33148,244,0,0,0,0;

%e .58786,1744314,10003422,16974314,9459090,1614906,61806,162,0,0,0,0;

%e .208012,8780912,71692452,180308420,161380816,51436848,5090124,103932,84,0,0,0,0;

%e ....

%e where the g.f. of row n is (1-x)^n*[g.f. of column n of A158825];

%e g.f. of row n of array A158825 is the n-th iteration of x*C(x):

%e .1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;

%e .1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;

%e .1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;

%e .1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;

%e .1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;

%e .1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;

%e .1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;

%e .1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...;

%e ....

%e ROW-REVERSAL yields triangle A122890:

%e .1;

%e .0,1;

%e .0,0,2;

%e .0,0,1,5;

%e .0,0,0,10,14;

%e .0,0,0,8,70,42;

%e .0,0,0,4,160,424,132;

%e .0,0,0,1,250,1978,2382,429;

%e .0,0,0,0,302,6276,19508,12804,1430; ...

%e where g.f. of row n = (1-x)^n*[g.f. of column n of A122888];

%e g.f. of row n of A122888 is the n-th iteration of x+x^2:

%e .1;

%e .1,1;

%e .1,2,2,1;

%e .1,3,6,9,10,8,4,1;

%e .1,4,12,30,64,118,188,258,302,298,244,162,84,32,8,1; ...

%t nmax = 11;

%t f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand;

%t T = Table[SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];

%t row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x] // Reverse;

%t Table[row[n], {n, 1, nmax}] // Flatten (* _Jean-François Alcover_, Oct 26 2018 *)

%o (PARI) {T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P); M=matrix(n+2, n+2, r, c, F=x; for(i=1, r, F=subst(F, x, CAT)); polcoeff(F, c)); Vec(truncate(Ser(vector(n+1,r,M[r,n+1])))*(1-x)^(n+1) +x*O(x^k))[k+1]}

%Y Cf. A158825, A122890 (row-reversal), A122888, columns: A000108, A122892.

%K nonn,tabl

%O 1,4

%A _Paul D. Hanna_, Mar 28 2009

%E Edited by _N. J. A. Sloane_, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.