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A132870
Triangle T, read by rows, where the g.f. of row n of T^n = (n^2 + y)^n for 0 <= n <= 29, where T^n denotes the n-th power of T considered as (lower-left) matrix.
11
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
OFFSET
0,4
COMMENTS
Pascal's triangle, C, obeys: g.f. of row n of C^n = (n + y)^n for n >= 0.
Starting from row 30 on, the terms computed by the given formulae are not integers any more (as noticed by Alois P. Heinz). - M. F. Hasler, Nov 19 2017
LINKS
M. F. Hasler, Table of n, a(n) for n = 0..464 (Rows 0..29; from row 30 on the table would contain fractions.)
FORMULA
Diagonals: T(n,n) = 1 for n >= 0. T(n,n-1) = n^2 for n >= 1. T(n, n-2) = (n-1)^2*n*(2n-1)/2. - M. F. Hasler, Nov 19 2017
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 (padded with 0's to the right of the diagonal) begins:
1;
2, 1;
16, 8, 1; <== g.f. of row 2: (2^2 + y)^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: (3^2 + y)^3
37380, 9048, 936, 48, 1;
3534210, 614925, 51000, 2550, 75, 1; ...
Matrix 4th power T^4 begins:
1;
4, 1;
48, 16, 1;
1266, 396, 36, 1;
65536, 16384, 1536, 64, 1; <== g.f. of row 4: (4^2 + y)^4
6058330, 1142800, 94100, 4200, 100, 1; ...
Matrix 5th power T^5 begins:
1;
5, 1;
70, 20, 1;
2010, 585, 45, 1;
106740, 26840, 2280, 80, 1;
9765625, 1953125, 156250, 6250, 125, 1; <== g.f. of row 5: (5^2 + y)^5
1431275130, 222982380, 15380550, 601500, 13950, 180, 1; ...
etc.
MATHEMATICA
pt = {{1}}; Table[rhs = CoefficientList[(k^2 + x)^k, x];
qt = Join[pt, {vars = Array[Subscript[a, #] &, k + 1]}];
b = MatrixPower[PadRight[qt], k] ;
{out} = vars /. Solve[Thread[Reverse[Last[b]] == Reverse[rhs]], vars];
pt = Join[pt, {out}]; out, {k, 20}] (* Wouter Meeussen, Nov 18 2017 *)
PROG
(PARI) T(n, k=-1/*omit 2nd arg to get the whole table up to row n*/, M=Mat(1))={ for(m=#M, n, M=matid(m+1)-matconcat([M^m; Vecrev(('x+m^2)^m)]); M=sum(i=1, m+1, -M^i/i)/m; M=sum(i=0, m+1, M^i/i!)); if(k>=0, M[n+1, k+1], M)}\\ Rewritten by M. F. Hasler, Nov 19 2017
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
A=T(29); c=-1; for(n=0, 29, for(k=0, n, write("/tmp/b132870.txt", c++, " ", A[n+1, k+1]))) \\ M. F. Hasler, Nov 18 2017
CROSSREFS
Cf. A132871 (row sums), columns: A132872, A132873, A132874.
Cf. variants: A177390, A132875.
Sequence in context: A278071 A362191 A362202 * A117254 A211022 A021613
KEYWORD
nonn,tabl,fini,full
AUTHOR
Paul D. Hanna, Sep 29 2007
EXTENSIONS
Edits made for clarification by Paul D. Hanna, May 25 2010
Further edits by M. F. Hasler, Nov 19 2017
STATUS
approved