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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).
12
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, 15712342, 5074738, 906557, 116372, 12035, 1082, 91, 8, 1, 357459042, 113775490, 19910808, 2483706
OFFSET
1,4
COMMENTS
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
EXAMPLE
Triangle T begins:
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;
15712342,5074738,906557,116372,12035,1082,91,8,1;
357459042,113775490,19910808,2483706,246596,20859,1589,116,9,1;
9094926988,2861365660,492818850,60168736,5801510,470928,33747,2232,144,10,1;
...
Array A158825 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,...;
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,...;
1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...;
1,9,90,945,10230,113190,1273668,14528217,167607066,...;
1,10,110,1265,14960,180510,2212188,27454218,344320262,...;
...
This triangle transforms diagonals of A158825 into each other:
where:
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,...].
PROG
(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]}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
Cf. columns: A158836, A158837, A158838, A158839, row sums: A158840.
Sequence in context: A152406 A105623 A364870 * A236961 A245958 A138271
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 28 2009, Mar 29 2009
EXTENSIONS
Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.
STATUS
approved