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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
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.
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Feb 12 2010
STATUS
approved