A173220 Triangle L, read by rows, equal to the matrix log of triangle A173210.

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.

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