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
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
Programs
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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]}
Comments