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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).

%I A166905 #2 Mar 30 2012 18:37:18
%S A166905 1,1,1,6,4,1,54,33,9,1,640,380,108,16,1,9380,5510,1610,270,25,1,
%T A166905 163576,95732,28560,5148,570,36,1,3305484,1933288,586320,110929,13650,
%U A166905 1071,49,1,75915708,44437080,13658904,2677008,353600,31624,1848,64,1,1952409954
%N 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).
%e A166905 Triangle begins:
%e A166905 1;
%e A166905 1,1;
%e A166905 6,4,1;
%e A166905 54,33,9,1;
%e A166905 640,380,108,16,1;
%e A166905 9380,5510,1610,270,25,1;
%e A166905 163576,95732,28560,5148,570,36,1;
%e A166905 3305484,1933288,586320,110929,13650,1071,49,1;
%e A166905 75915708,44437080,13658904,2677008,353600,31624,1848,64,1;
%e A166905 1952409954,1144564278,355787568,71648322,9962949,973845,66150,2988,81,1;
%e A166905 55573310936,32638644236,10243342296,2107966432,304857190,31795560,2395120,127720,4590,100,1;
%e A166905 ...
%e A166905 Coefficients in iterations of x*Catalan(x) form table A158825:
%e A166905 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
%e A166905 1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
%e A166905 1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
%e A166905 1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
%e A166905 1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
%e A166905 1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
%e A166905 1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
%e A166905 ...
%e A166905 This triangle T transforms rows into diagonals of A158825;
%e A166905 the initial diagonals begin:
%e A166905 A158831: [1,1,6,54,640,9380,163576,3305484,...];
%e A166905 A158832: [1,2,12,110,1330,19852,351792,7209036,...];
%e A166905 A158833: [1,3,20,195,2464,38052,693048,14528217,...];
%e A166905 A158834: [1,4,30,315,4200,67620,1273668,27454218,...].
%e A166905 For example:
%e A166905 T * [1,0,0,0,0,0,0,0,0,0,0,0,0, ...] = A158831;
%e A166905 T * [1,1,2,5,14,42,132,429,1430,...] = A158832;
%e A166905 T * [1,2,6,21,80,322,1348,5814, ...] = A158833;
%e A166905 T * [1,3,12,54,260, 1310, 6824, ...] = A158834.
%o A166905 (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]}
%Y A166905 Cf. A166906, A166907, A166908, A166909, variant: A166900.
%Y A166905 Cf. A158825, A158831, A158832, A158833, A158834, A158835, A000108.
%K A166905 nonn,tabl
%O A166905 0,4
%A A166905 _Paul D. Hanna_, Nov 28 2009