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A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

%I A158835 #17 Oct 16 2024 21:52:56
%S A158835 1,1,1,4,2,1,27,11,3,1,254,94,21,4,1,3062,1072,217,34,5,1,45052,15212,
%T A158835 2904,412,50,6,1,783151,257777,47337,6325,695,69,7,1,15712342,5074738,
%U A158835 906557,116372,12035,1082,91,8,1,357459042,113775490,19910808,2483706
%N A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).
%C A158835 Conjecture: n-th reversed row polynomial is t_n where we start with vector v of fixed length m with elements v_i = 1, then set t := v and for i=1..m-1, for j=1..i, for k=j+1..i+1 apply v_k := v_k + z*v_{k-1} and t_{i+1} := v_{i+1} (after ending each cycle for j). - _Mikhail Kurkov_, Sep 03 2024
%H A158835 Paul D. Hanna, <a href="/A158835/b158835.txt">Table of n, a(n), n = 1..496 (rows 1..31).</a>
%e A158835 Triangle T begins:
%e A158835   1;
%e A158835   1,1;
%e A158835   4,2,1;
%e A158835   27,11,3,1;
%e A158835   254,94,21,4,1;
%e A158835   3062,1072,217,34,5,1;
%e A158835   45052,15212,2904,412,50,6,1;
%e A158835   783151,257777,47337,6325,695,69,7,1;
%e A158835   15712342,5074738,906557,116372,12035,1082,91,8,1;
%e A158835   357459042,113775490,19910808,2483706,246596,20859,1589,116,9,1;
%e A158835   9094926988,2861365660,492818850,60168736,5801510,470928,33747,2232,144,10,1;
%e A158835   ...
%e A158835 Array A158825 of coefficients in iterations of x*C(x) begins:
%e A158835   1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
%e A158835   1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
%e A158835   1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
%e A158835   1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
%e A158835   1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
%e A158835   1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
%e A158835   1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
%e A158835   1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...;
%e A158835   1,9,90,945,10230,113190,1273668,14528217,167607066,...;
%e A158835   1,10,110,1265,14960,180510,2212188,27454218,344320262,...;
%e A158835   ...
%e A158835 This triangle transforms diagonals of A158825 into each other:
%e A158835 T*A158831 = A158832; T*A158832 = A158833; T*A158833 = A158834;
%e A158835 where:
%e A158835 A158831 = [1,1,6,54,640,9380,163576,3305484,...];
%e A158835 A158832 = [1,2,12,110,1330,19852,351792,7209036,...];
%e A158835 A158833 = [1,3,20,195,2464,38052,693048,14528217,...];
%e A158835 A158834 = [1,4,30,315,4200,67620,1273668,27454218,...].
%o A158835 (PARI) {T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%o A158835 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A158835 Cf. columns: A158836, A158837, A158838, A158839, row sums: A158840.
%Y A158835 Cf. A158825, A158831, A158832, A158833, A158834, variant: A135080.
%K A158835 nonn,tabl
%O A158835 1,4
%A A158835 _Paul D. Hanna_, Mar 28 2009, Mar 29 2009
%E A158835 Edited by _N. J. A. Sloane_, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.