A132875 Triangle T, read by rows, where g.f. of row n of T^n = (y + n*(n+1))^n for n>=0 and T^n denotes the n-th matrix power of T.

1, 2, 1, 12, 6, 1, 240, 72, 12, 1, 12400, 2240, 240, 20, 1, 1242720, 157200, 10800, 600, 30, 1, 202721064, 20017032, 1000440, 36960, 1260, 42, 1, 48537596352, 3986643136, 159475008, 4390400, 101920, 2352, 56, 1, 15957585674496, 1133413590528

Offset

0,2

Comments

Pascal's triangle, B, obeys: the g.f. of row n of B^n = (y + n)^n for n>=0; this triangle has a similar property.

Links

Table of n, a(n) for n=0..37.

Formula

C(n,k) divides T(n,k) for n>=k>=0.

Example

Triangle begins:
1;
2, 1;
12, 6, 1;
240, 72, 12, 1;
12400, 2240, 240, 20, 1;
1242720, 157200, 10800, 600, 30, 1;
202721064, 20017032, 1000440, 36960, 1260, 42, 1;
48537596352, 3986643136, 159475008, 4390400, 101920, 2352, 56, 1;
15957585674496, 1133413590528, 38423427840, 860840064, 15140160, 241920, 4032, 72, 1; ...
Matrix square T^2 begins:
1;
4, 1;
36, 12, 1; <== g.f. of row 2: (y + 2*3)^2
768, 216, 24, 1;
36960, 7360, 720, 40, 1;
3445440, 489600, 36000, 1800, 60, 1; ...
Matrix cube T^3 begins:
1;
6, 1;
72, 18, 1;
1728, 432, 36, 1; <== g.f. of row 3: (y + 3*4)^3
82320, 16800, 1440, 60, 1;
7275360, 1126800, 82800, 3600, 90, 1; ...
Matrix 4th power T^4 begins:
1;
8, 1;
120, 24, 1;
3264, 720, 48, 1;
160000, 32000, 2400, 80, 1; <== g.f. of row 4: (y + 4*5)^4
13745280, 2241600, 158400, 6000, 120, 1; ...

Prog

(PARI) {T(n, k)=local(M=Mat(1), N, L); for(i=1, n, N=M; M=matrix(#N+1, #N+1, r, c, if(r>=c, if(r<=#N, (N^(#N))[r, c], polcoeff((x+(#M)*(#M+1))^(#M), c-1)))); 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. A132876 (row sums); columns: A132877, A132878; A132879; variant: A132870.

Sequence in context: A008285 A119274 A066991 * A050139 A010255 A372508

Adjacent sequences: A132872 A132873 A132874 * A132876 A132877 A132878

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 29 2007

Status

approved