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.

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

Adjacent sequences: A132867 A132868 A132869 * A132871 A132872 A132873

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