A172091 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 = 2.

1, 1, 1, 1, -1, 1, 1, -17, -17, 1, 1, -293, -309, -293, 1, 1, -9449, -9741, -9741, -9449, 1, 1, -605429, -614877, -615153, -614877, -605429, 1, 1, -77514569, -78119997, -78129429, -78129429, -78119997, -77514569, 1, 1, -19844960309, -19922474877, -19923080289, -19923089445, -19923080289, -19922474877, -19844960309, 1

Offset

0,8

Comments

Row sums are: {1, 2, 1, -32, -893, -38378, -3055763, -467527988, -139304120393,

-81405588536318, -93713294552041343, ...}.

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=2.

Example

Triangle begins as:
  1;
  1,       1;
  1,      -1,       1;
  1,     -17,     -17,       1;
  1,    -293,    -309,    -293,       1;
  1,   -9449,   -9741,   -9741,   -9449,       1;
  1, -605429, -614877, -615153, -614877, -605429, 1;

Maple

T:= proc(n, k, q) option remember;
    c(n, q):= mul( 2^(j+1) -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, 2), k=0..n), n=0..10); # G. C. Greubel, Dec 05 2019

Mathematica

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

Crossrefs

Cf. this sequence (q=2), A172092 (q=3), A172093 (q=4).

Sequence in context: A082123 A368929 A273973 * A371997 A291370 A291432

Adjacent sequences: A172088 A172089 A172090 * A172092 A172093 A172094

Keyword

sign,tabl

Author

Roger L. Bagula, Jan 25 2010

Extensions

Edited by G. C. Greubel, Dec 05 2019

Status

approved