A172093 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 = 4.

1, 1, 1, 1, -3, 1, 1, -99, -99, 1, 1, -8819, -8915, -8819, 1, 1, -3034499, -3043315, -3043315, -3034499, 1, 1, -4151231699, -4154266195, -4154274915, -4154266195, -4151231699, 1, 1, -22682342182499, -22686493414195, -22686496448595, -22686496448595, -22686493414195, -22682342182499, 1

Offset

0,5

Comments

Row sums are: {1, 2, -2, -356, -86660, -90587564, -442533635468, -10372635857431772, -1181791865462943686876, ...}.

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=4.

Example

Triangle begins as:
  1;
  1,        1;
  1,       -3,        1;
  1,      -99,      -99,        1;
  1,    -8819,    -8915,    -8819,        1;
  1, -3034499, -3043315, -3043315, -3034499, 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, 4), 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, 4], {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, 4), ", "))) \\ 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, 4): 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, 4) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 05 2019

Crossrefs

Cf. A172091 (q=2), A172092 (q=3), this sequence (q=4).

Sequence in context: A168549 A010272 A368026 * A294402 A145738 A232802

Adjacent sequences: A172090 A172091 A172092 * A172094 A172095 A172096

Keyword

sign,tabl

Author

Roger L. Bagula, Jan 25 2010

Extensions

Edited by G. C. Greubel, Dec 05 2019

Status

approved