A172092 Triangle, read by rows, T(n,k,q) = c(k,q) + c(n-k,q) - c(n, q) where c(n,q) = Product_{j=1..n-1} ((q^(j+1) - 1)/(q-1)) and q = 3.

1, 1, 1, 1, -2, 1, 1, -47, -47, 1, 1, -2027, -2072, -2027, 1, 1, -249599, -251624, -251624, -249599, 1, 1, -91359839, -91609436, -91611416, -91609436, -91359839, 1, 1, -100039779839, -100131139676, -100131389228, -100131389228, -100131139676, -100039779839, 1

Offset

0,5

Comments

Row sums are: {1, 2, 0, -92, -6124, -1002444, -457549964, -600604617484, -2298816299112204, -25856055844713627404, -858811326017167374184204, ...}.

Links

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Formula

Let c(n,q) = Product_{j=1..n-1} ((q^(j+1) - 1)/(q-1)) then T(n,k,q) = -c(n,q) + c(n-k,q) + c(k, q) for q=3.

Example

Triangle begins as:
  1;
  1,         1;
  1,        -2,         1;
  1,       -47,       -47,         1;
  1,     -2027,     -2072,     -2027,         1;
  1,   -249599,   -251624,   -251624,   -249599,         1;
  1, -91359839, -91609436, -91611416, -91609436, -91359839, 1;

Maple

T:= proc(n, k, q) option remember;
    c(n, q):= mul( (q^(j+1) -1)/(q-1), j=1..n-1);
    T(n, k, q):= c(k, q) + c(n-k, q) - c(n, q);
   end:
seq(seq(T(n, k, 3), k=0..n), n=0..10); # G. C. Greubel, Dec 05 2019

Mathematica

c[n_, q_]:= Product[(q^(j+1) -1)/(q-1), {j, n-1}]; T[n_, k_, q_]:= c[k, q] + c[n-k, q] - c[n, q]; Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Dec 05 2019 *)

Prog

(PARI) c(n, q) = prod(j=1, n-1, (q^(j+1) -1)/(q-1));
T(n, k, q) = c(k, q) + c(n-k, q) - c(n, q);
for(n=0, 10, for(k=0, n, print1(T(n, k, 3), ", "))) \\ G. C. Greubel, Dec 05 2019
(Magma) c:= func< n, q | n lt 2 select 1 else &*[(q^(j+1) -1)/(q-1): j in [1..n-1]] >;
T:= func< n, k, q | c(k, q) + c(n-k, q) - c(n, q) >;
[T(n, k, 3): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 05 2019
(Sage)
def c(n, q): return product( (q^(j+1) -1)/(q-1) for j in (1..n-1))
def T(n, k, q): return c(k, q) + c(n-k, q) - c(n, q)
[[T(n, k, 3) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 05 2019

Crossrefs

Cf. A172091 (q=2), this sequence (q=3), A172093 (q=4).

Sequence in context: A182911 A362226 A058293 * A156888 A173890 A159767

Adjacent sequences: A172089 A172090 A172091 * A172093 A172094 A172095

Keyword

sign,tabl

Author

Roger L. Bagula, Jan 25 2010

Extensions

Edited by G. C. Greubel, Dec 05 2019

Status

approved