editing

approved

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

approved

editing

Edited by _N. J. A. Sloane, _, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.

OEIS Server: https://oeis.org/edit/global/1903

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Mar 28 2009, Mar 29 2009

OEIS Server: https://oeis.org/edit/global/213

Paul D. Hanna, <a href="="/A158835/b158835.txt">Table of n, a(n), n = 1..496 (rows 1..31).</a>

nonn,tabl,new

Paul D. Hanna, <a href="b158835.txt">Table of n, a(n), n = 01..495496 (rows 01..3031).</a>

Triangle T begins: 1;:

nonn,tabl,new

Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.

Paul D. Hanna, <a href="b158835.txt">Table of n, a(n), n = 0..495 (rows 0..30).</a>

nonn,new

nonn

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).

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

1,4

Triangle T begins: 1;

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:

T*A158831 = A158832; T*A158832 = A158833; T*A158833 = A158834;

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,...].

(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]}

Cf. columns: A158836, A158837, A158838, A158839, row sums: A158840.

Cf. A158825, A158831, A158832, A158833, A158834, variant: A135080.

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 28 2009, Mar 29 2009

approved