OFFSET
0,2
COMMENTS
Pascal's triangle, B, obeys: the g.f. of row n of B^n = (y + n)^n for n>=0; this triangle has a similar property.
FORMULA
C(n,k) divides T(n,k) for n>=k>=0.
EXAMPLE
Triangle begins:
1;
2, 1;
12, 6, 1;
240, 72, 12, 1;
12400, 2240, 240, 20, 1;
1242720, 157200, 10800, 600, 30, 1;
202721064, 20017032, 1000440, 36960, 1260, 42, 1;
48537596352, 3986643136, 159475008, 4390400, 101920, 2352, 56, 1;
15957585674496, 1133413590528, 38423427840, 860840064, 15140160, 241920, 4032, 72, 1; ...
Matrix square T^2 begins:
1;
4, 1;
36, 12, 1; <== g.f. of row 2: (y + 2*3)^2
768, 216, 24, 1;
36960, 7360, 720, 40, 1;
3445440, 489600, 36000, 1800, 60, 1; ...
Matrix cube T^3 begins:
1;
6, 1;
72, 18, 1;
1728, 432, 36, 1; <== g.f. of row 3: (y + 3*4)^3
82320, 16800, 1440, 60, 1;
7275360, 1126800, 82800, 3600, 90, 1; ...
Matrix 4th power T^4 begins:
1;
8, 1;
120, 24, 1;
3264, 720, 48, 1;
160000, 32000, 2400, 80, 1; <== g.f. of row 4: (y + 4*5)^4
13745280, 2241600, 158400, 6000, 120, 1; ...
PROG
(PARI) {T(n, k)=local(M=Mat(1), N, L); for(i=1, n, N=M; M=matrix(#N+1, #N+1, r, c, if(r>=c, if(r<=#N, (N^(#N))[r, c], polcoeff((x+(#M)*(#M+1))^(#M), c-1)))); L=sum(i=1, #M, -(M^0-M)^i/i); M=sum(i=0, #M, (L/#N)^i/i!); ); M[n+1, k+1]}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Sep 29 2007
STATUS
approved