A173210 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.

1, 1, 1, 4, 4, 1, 84, 36, 9, 1, 4584, 1056, 144, 16, 1, 469440, 73200, 6000, 400, 25, 1, 76982940, 9179640, 537300, 22800, 900, 36, 1, 18391183020, 1794887640, 83163780, 2598960, 67620, 1764, 49, 1, 6011375932800, 500614248960, 19475406720

Offset

0,4

Links

Table of n, a(n) for n=0..38.

Formula

[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]

Example

Triangle T begins:
1;
1,1;
4,4,1;
84,36,9,1;
4584,1056,144,16,1;
469440,73200,6000,400,25,1;
76982940,9179640,537300,22800,900,36,1;
18391183020,1794887640,83163780,2598960,67620,1764,49,1;
6011375932800,500614248960,19475406720,492038400,9619680,169344,3136,64,1;
2570927357779200,187826632116480,6361668149760,137161382400,2198871360,29502144,374976,5184,81,1;
...
Generator triangle I+D, with diagonal D(n+1,n)=n+1, begins:
1;
1,1;
0,2,1;
0,0,3,1;
0,0,0,4,1;
...
ILLUSTRATE: row n of T^n = row n of (I+D)^(n^2).
For n=2, matrix square T^2 begins:
1;
2,1;
12,8,1; <== row 2 of T^2
240,108,18,1;
12144,3264,432,32,1;
...
while (I+D)^4 begins:
1;
4,1;
12,8,1; <== row 2 of (I+D)^4 = row 2 of T^2
24,36,12,1;
24,96,72,16,1;
...
For n=3, matrix cube T^3 begins:
1;
3,1;
24,12,1;
504,216,27,1; <== row 3 of T^3
24408,7200,864,48,1;
...
while (I+D)^9 begins:
1;
9,1;
72,18,1;
504,216,27,1; <== row 3 of (I+D)^9 = row 3 of T^3
3024,2016,432,36,1;
...
For n=4, matrix power T^4 begins:
1;
4,1;
40,16,1;
912,360,36,1;
43680,13440,1440,64,1; <== row 4 of T^4
...
while (I+D)^16 begins:
1;
16,1;
240,32,1;
3360,720,48,1;
43680,13440,1440,64,1; <== row 4 of (I+D)^16 = row 4 of T^4
...

Prog

(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]}

Crossrefs

Cf. columns: A173211, A173212, A173213, variant: A132870.

Sequence in context: A111845 A120396 A141024 * A328922 A058888 A194678

Adjacent sequences: A173207 A173208 A173209 * A173211 A173212 A173213

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 12 2010

Status

approved