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A136214 Triangle U, read by rows, where U(n,k) = Product_{j=k..n-1} (3*j+1) for n > k with U(n,n) = 1. 7
1, 1, 1, 4, 4, 1, 28, 28, 7, 1, 280, 280, 70, 10, 1, 3640, 3640, 910, 130, 13, 1, 58240, 58240, 14560, 2080, 208, 16, 1, 1106560, 1106560, 276640, 39520, 3952, 304, 19, 1, 24344320, 24344320, 6086080, 869440, 86944, 6688, 418, 22, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) = A112292(n, k) and T(n, k, 3) is this sequence. - Peter Luschny, Jun 01 2009, revised Jun 18 2019
LINKS
FORMULA
Matrix powers: column 0 of U^(k+1) = column k of A136216 for k >= 0; simultaneously, column k = column 0 of A136216^(3k+1) for k >= 0. Element in column 0, row n, of matrix power U^(k+1) = A007559(n)*C(n+k,k), where A007559 are triple factorials found in column 0 of this triangle.
EXAMPLE
Triangle begins:
1;
1, 1;
4, 4, 1;
28, 28, 7, 1;
280, 280, 70, 10, 1;
3640, 3640, 910, 130, 13, 1;
58240, 58240, 14560, 2080, 208, 16, 1;
1106560, 1106560, 276640, 39520, 3952, 304, 19, 1; ...
Matrix inverse begins:
1;
-1, 1;
0, -4, 1;
0, 0, -7, 1;
0, 0, 0, -10, 1;
0, 0, 0, 0, -13, 1; ...
MAPLE
nmax:=8; for n from 0 to nmax do U(n, n):=1 od: for n from 0 to nmax do for k from 0 to n do if n > k then U(n, k) := mul((3*j+1), j = k..n-1) fi: od: od: for n from 0 to nmax do seq(U(n, k), k=0..n) od: seq(seq(U(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jul 04 2011, revised Nov 23 2012
MATHEMATICA
Table[Product[3*j+1, {j, k, n-1}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2019 *)
PROG
(PARI) T(n, k)=if(n==k, 1, prod(j=k, n-1, 3*j+1))
(Magma) [[n eq 0 select 1 else k eq n select 1 else (&*[3*j+1: j in [k..n-1]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 14 2019
(Sage)
def T(n, k):
if (k==n): return 1
else: return product(3*j+1 for j in (k..n-1))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 14 2019
CROSSREFS
Sequence in context: A126280 A170986 A071207 * A067328 A111845 A120396
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Feb 07 2008
STATUS
approved

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Last modified April 19 17:51 EDT 2024. Contains 371797 sequences. (Running on oeis4.)