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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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A058888 A194678 A153015

Adjacent sequences:  A173207 A173208 A173209 * A173211 A173212 A173213

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Feb 12 2010

STATUS

approved

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Last modified August 25 02:56 EDT 2019. Contains 326318 sequences. (Running on oeis4.)