A166905 Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

1, 1, 1, 6, 4, 1, 54, 33, 9, 1, 640, 380, 108, 16, 1, 9380, 5510, 1610, 270, 25, 1, 163576, 95732, 28560, 5148, 570, 36, 1, 3305484, 1933288, 586320, 110929, 13650, 1071, 49, 1, 75915708, 44437080, 13658904, 2677008, 353600, 31624, 1848, 64, 1, 1952409954

Offset: 0

Author

Paul D. Hanna, Nov 28 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1;
1,1;
6,4,1;
54,33,9,1;
640,380,108,16,1;
9380,5510,1610,270,25,1;
163576,95732,28560,5148,570,36,1;
3305484,1933288,586320,110929,13650,1071,49,1;
75915708,44437080,13658904,2677008,353600,31624,1848,64,1;
1952409954,1144564278,355787568,71648322,9962949,973845,66150,2988,81,1;
55573310936,32638644236,10243342296,2107966432,304857190,31795560,2395120,127720,4590,100,1;
...
Coefficients in iterations of x*Catalan(x) form table A158825:
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,...;
...
This triangle T transforms rows into diagonals of A158825;
the initial diagonals begin:
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,...].
For example:
T * [1,0,0,0,0,0,0,0,0,0,0,0,0, ...] = A158831;
T * [1,1,2,5,14,42,132,429,1430,...] = A158832;
T * [1,2,6,21,80,322,1348,5814, ...] = A158833;
T * [1,3,12,54,260, 1310, 6824, ...] = A158834.
		

Crossrefs

Cf. A166906, A166907, A166908, A166909, variant: A166900.

Cf. A158825, A158831, A158832, A158833, A158834, A158835, A000108.

Programs

PARI

{T(n, k)=local(F=x, G=serreverse(x-x^2+O(x^(n+3))), M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x;for(i=1, r, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}