A173210 - OEIS

%I #2 Mar 30 2012 18:37:21

%S 1,1,1,4,4,1,84,36,9,1,4584,1056,144,16,1,469440,73200,6000,400,25,1,

%T 76982940,9179640,537300,22800,900,36,1,18391183020,1794887640,

%U 83163780,2598960,67620,1764,49,1,6011375932800,500614248960,19475406720

%N Triangle T such that row n of T^n = row n of (I+D)^(n^2) where D is the lower diagonal matrix: D(n+1,n)=n+1, and I is the identity matrix.

%F [T^n](n,k) = (n-k)!*C(n,k)*C(n^2,n-k) gives the k-th term of the n-th row of the n-th matrix power of this triangle T. [From _Paul D. Hanna_, Feb 13 2010]

%e Triangle T begins:

%e 1;

%e 1,1;

%e 4,4,1;

%e 84,36,9,1;

%e 4584,1056,144,16,1;

%e 469440,73200,6000,400,25,1;

%e 76982940,9179640,537300,22800,900,36,1;

%e 18391183020,1794887640,83163780,2598960,67620,1764,49,1;

%e 6011375932800,500614248960,19475406720,492038400,9619680,169344,3136,64,1;

%e 2570927357779200,187826632116480,6361668149760,137161382400,2198871360,29502144,374976,5184,81,1;

%e ...

%e Generator triangle I+D, with diagonal D(n+1,n)=n+1, begins:

%e 1;

%e 1,1;

%e 0,2,1;

%e 0,0,3,1;

%e 0,0,0,4,1;

%e ...

%e ILLUSTRATE: row n of T^n = row n of (I+D)^(n^2).

%e For n=2, matrix square T^2 begins:

%e 1;

%e 2,1;

%e 12,8,1; <== row 2 of T^2

%e 240,108,18,1;

%e 12144,3264,432,32,1;

%e ...

%e while (I+D)^4 begins:

%e 1;

%e 4,1;

%e 12,8,1; <== row 2 of (I+D)^4 = row 2 of T^2

%e 24,36,12,1;

%e 24,96,72,16,1;

%e ...

%e For n=3, matrix cube T^3 begins:

%e 1;

%e 3,1;

%e 24,12,1;

%e 504,216,27,1; <== row 3 of T^3

%e 24408,7200,864,48,1;

%e ...

%e while (I+D)^9 begins:

%e 1;

%e 9,1;

%e 72,18,1;

%e 504,216,27,1; <== row 3 of (I+D)^9 = row 3 of T^3

%e 3024,2016,432,36,1;

%e ...

%e For n=4, matrix power T^4 begins:

%e 1;

%e 4,1;

%e 40,16,1;

%e 912,360,36,1;

%e 43680,13440,1440,64,1; <== row 4 of T^4

%e ...

%e while (I+D)^16 begins:

%e 1;

%e 16,1;

%e 240,32,1;

%e 3360,720,48,1;

%e 43680,13440,1440,64,1; <== row 4 of (I+D)^16 = row 4 of T^4

%e ...

%o (PARI) {T(n, k)=local(M=Mat(1), N, L, C=matrix(n+1,n+1,r,c,if(r==c,1,if(r==c+1,c)))); for(i=1, n, N=M; M=matrix(#N+1, #N+1, r, c, if(r>=c, if(r<=#N, (N^(#N))[r, c], (C^((#M)^2))[r, c]))); 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. columns: A173211, A173212, A173213, variant: A132870.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Feb 12 2010