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A173220 Triangle L, read by rows, equal to the matrix log of triangle A173210. 5
0, 1, 0, 2, 4, 0, 60, 18, 9, 0, 3528, 672, 72, 16, 0, 377620, 51600, 3600, 200, 25, 0, 63642600, 6883920, 359100, 13200, 450, 36, 0, 15485843814, 1399989780, 59539410, 1675800, 38220, 882, 49, 0, 5128034120832, 401060653056, 14585598720 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Triangle T = A173210 satisfies: row n of T^n = row n of (I+D)^(n^2) for n>=0, where D is the lower diagonal matrix: D(n+1,n)=n+1.

LINKS

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

FORMULA

Matrix A173220 = Sum_{n>=1} -[I - A173210]^n/n.

Matrix A173210 = Sum_{n>=0} [A173220]^n/n!.

EXAMPLE

Matrix L begins:

0;

1,0;

2,4,0;

60,18,9,0;

3528,672,72,16,0;

377620,51600,3600,200,25,0;

63642600,6883920,359100,13200,450,36,0;

15485843814,1399989780,59539410,1675800,38220,882,49,0;

5128034120832,401060653056,14585598720,340820480,6044640,94080,1568,64,0;

2214330161840640,153389081849472,4915556440704,99679870080,1485857520,18180288,205632,2592,81,0;

...

Matrix exponentiation yields triangle T = A173210:

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;

...

where row n of T^n = row n of (I+D)^(n^2) and

matrix 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; ...

ILLUSTRATION of the definition of T = A173210:

at 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=Mat(1), C=matrix(n+2, n+2, r, c, if(r==c, 1, if(r==c+1, c)))); for(i=1, n+1, 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!); ); L[n+1, k+1]/(n+1)}

CROSSREFS

Cf. columns: A173221, A173222, A173223, A173224 (row sums), A173210.

Sequence in context: A347281 A055871 A012716 * A012362 A012710 A333367

Adjacent sequences:  A173217 A173218 A173219 * A173221 A173222 A173223

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Feb 12 2010

STATUS

approved

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Last modified June 26 03:55 EDT 2022. Contains 354871 sequences. (Running on oeis4.)