A136214 - OEIS

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.

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

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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

G. C. Greubel, Rows n = 0..100 of triangle, flattened

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;

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;

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

Cf. A094587, A112333, A136216, A136239; A007559, A136212, A136213.