A132875 - OEIS

%I #5 Mar 14 2015 11:36:22

%S 1,2,1,12,6,1,240,72,12,1,12400,2240,240,20,1,1242720,157200,10800,

%T 600,30,1,202721064,20017032,1000440,36960,1260,42,1,48537596352,

%U 3986643136,159475008,4390400,101920,2352,56,1,15957585674496,1133413590528

%N Triangle T, read by rows, where g.f. of row n of T^n = (y + n*(n+1))^n for n>=0 and T^n denotes the n-th matrix power of T.

%C 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.

%F C(n,k) divides T(n,k) for n>=k>=0.

%e Triangle begins:

%e 1;

%e 2, 1;

%e 12, 6, 1;

%e 240, 72, 12, 1;

%e 12400, 2240, 240, 20, 1;

%e 1242720, 157200, 10800, 600, 30, 1;

%e 202721064, 20017032, 1000440, 36960, 1260, 42, 1;

%e 48537596352, 3986643136, 159475008, 4390400, 101920, 2352, 56, 1;

%e 15957585674496, 1133413590528, 38423427840, 860840064, 15140160, 241920, 4032, 72, 1; ...

%e Matrix square T^2 begins:

%e 1;

%e 4, 1;

%e 36, 12, 1; <== g.f. of row 2: (y + 2*3)^2

%e 768, 216, 24, 1;

%e 36960, 7360, 720, 40, 1;

%e 3445440, 489600, 36000, 1800, 60, 1; ...

%e Matrix cube T^3 begins:

%e 1;

%e 6, 1;

%e 72, 18, 1;

%e 1728, 432, 36, 1; <== g.f. of row 3: (y + 3*4)^3

%e 82320, 16800, 1440, 60, 1;

%e 7275360, 1126800, 82800, 3600, 90, 1; ...

%e Matrix 4th power T^4 begins:

%e 1;

%e 8, 1;

%e 120, 24, 1;

%e 3264, 720, 48, 1;

%e 160000, 32000, 2400, 80, 1; <== g.f. of row 4: (y + 4*5)^4

%e 13745280, 2241600, 158400, 6000, 120, 1; ...

%o (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]}

%Y Cf. A132876 (row sums); columns: A132877, A132878; A132879; variant: A132870.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Sep 29 2007