1,4

Row sums equal the factorial numbers.

...

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.

...

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.

Triangle begins:

.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;

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

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

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

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

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

....

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

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

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

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

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

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

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

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

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

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

....

ROW-REVERSAL yields triangle A122890:

.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; ...

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

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

.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; ...

(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]}

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

Sequence in context: A057611 A147843 A094597 this_sequence A145430 A143160 A156387

Adjacent sequences: A158827 A158828 A158829 this_sequence A158831 A158832 A158833

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 28 2009