1,5

G.f. of column n = [g.f. of row n of A158830]/(1-x)^n.

Row k equals the first column of the k-th matrix power of Catalan triangle A033184; thus triangle A033184 transforms row n into row n+1 of this array (A158825). [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2009]

Square array of coefficients in iterations of x*C(x) begins:

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

1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,11485068,...;

1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,35520498,...;

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

1,5,30,195,1330,9380,67844,500619,3755156,28558484,219767968,...;

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,...;

1,9,90,945,10230,113190,1273668,14528217,167607066,1952409954,...;

1,10,110,1265,14960,180510,2212188,27454218,344320262,...;

1,11,132,1650,21164,276562,3666520,49181418,666200106,...;

1,12,156,2106,29120,409682,5841836,84218134,1225314662,...;

1,13,182,2639,39130,589680,8999172,138755799,2157976392,...;

1,14,210,3255,51520,827960,13464752,221101608,3660331064,...;

1,15,240,3960,66640,1137640,19640032,342179672,6007747368,...;

1,16,272,4760,84864,1533672,28012464,516105720,9578580504,...; ...

ILLUSTRATE ITERATIONS.

Let G(x) = x*C(x), then the first few iterations of G(x) are:

G(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 +...

G(G(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 +...

G(G(G(x))) = x + 3*x^2 + 12*x^3 + 54*x^4 + 260*x^5 +...

G(G(G(G(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 640*x^5 +...

...

RELATED TRIANGLES.

The g.f. of column n is [g.f. of row n of A158830]/(1-x)^n

where triangle A158830 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;

...

Triangle A158835 transforms one diagonal into the next:

1;

1,1;

4,2,1;

27,11,3,1;

254,94,21,4,1;

3062,1072,217,34,5,1;

45052,15212,2904,412,50,6,1;

783151,257777,47337,6325,695,69,7,1; ...

so that:

A158835 * A158831 = A158832;

A158835 * A158832 = A158833;

A158835 * A158833 = A158834;

where the diagonals start:

A158831 = [1,1,6,54,640,9380,163576,3305484,...];

A158832 = [1,2,12,110,1330,19852,351792,7209036,...];

A158833 = [1,3,20,195,2464,38052,693048,14528217,...];

A158834 = [1,4,30,315,4200,67620,1273668,27454218,...].

(PARI) {T(n, k)=local(F=serreverse(x-x^2+O(x^(k+2))), G=x); for(i=1, n, G=subst(F, x, G)); polcoeff(G, k)}

Cf. rows: A000108, A121988, A158826, A158827, A158828; antidiagonal sums: A158829.

Cf. diagonals: A158831, A158832, A158833, A158834.

Cf. related triangles: A158830, A158835, variant: A122888.

Sequence in context: A098474 A153199 A056860 this_sequence A107111 A082037 A163649

Adjacent sequences: A158822 A158823 A158824 this_sequence A158826 A158827 A158828

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 28 2009, Mar 29 2009