|
Search: id:A132870
|
|
|
| A132870 |
|
Triangle T, read by rows, where the g.f. of row n of T^n = (y + n^2)^n for n>=0, where T^n denotes the matrix power n of T. |
|
+0
7
|
|
| 1, 1, 1, 6, 4, 1, 132, 45, 9, 1, 7156, 1432, 168, 16, 1, 729895, 101725, 7550, 450, 25, 1, 119636226, 12938076, 697590, 27420, 990, 36, 1, 28619359629, 2559100705, 110137692, 3226895, 78890, 1911, 49, 1, 9374688646296, 721024536688
(list; table; graph; listen)
|
|
|
OFFSET
|
0,4
|
|
|
COMMENT
|
Pascal's triangle, B, obeys: g.f. of row n of B^n = (y + n)^n for n>=0.
|
|
EXAMPLE
|
Triangle T begins:
1;
1, 1;
6, 4, 1;
132, 45, 9, 1;
7156, 1432, 168, 16, 1;
729895, 101725, 7550, 450, 25, 1;
119636226, 12938076, 697590, 27420, 990, 36, 1;
28619359629, 2559100705, 110137692, 3226895, 78890, 1911, 49, 1;
9374688646296, 721024536688, 26208036736, 624158528, 11572400, 193312, 3360, 64, 1; ...
Matrix square T^2 begins:
1;
2, 1;
16, 8, 1; <== g.f. of row 2: (y + 2^2)^2
363, 126, 18, 1;
18864, 4256, 480, 32, 1;
1845115, 289700, 23350, 1300, 50, 1; ...
Matrix cube T^3 begins:
1;
3, 1;
30, 12, 1;
729, 243, 27, 1; <== g.f. of row 3: (y + 3^2)^3
37380, 9048, 936, 48, 1;
3534210, 614925, 51000, 2550, 75, 1; ...
Matrix 4-th power T^4 begins:
1;
4, 1;
48, 16, 1;
1266, 396, 36, 1;
65536, 16384, 1536, 64, 1; <== g.f. of row 4: (y + 4^2)^4
6058330, 1142800, 94100, 4200, 100, 1; ...
|
|
PROGRAM
|
(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)^2)^(#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. A132871 (row sums), columns: A132872, A132873; A132874.
Sequence in context: A117335 A021863 A086241 this_sequence A117254 A021613 A073449
Adjacent sequences: A132867 A132868 A132869 this_sequence A132871 A132872 A132873
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Sep 29 2007
|
|
|
Search completed in 0.002 seconds
|
