A134319 Triangle read by rows. T(n, k) = binomial(n, k)*(2^k - 1 + 0^k).

1, 1, 1, 1, 2, 3, 1, 3, 9, 7, 1, 4, 18, 28, 15, 1, 5, 30, 70, 75, 31, 1, 6, 45, 140, 225, 186, 63, 1, 7, 63, 245, 525, 651, 441, 127, 1, 8, 84, 392, 1050, 1736, 1764, 1016, 255, 1, 9, 108, 588, 1890, 3906, 5292, 4572, 2295, 511, 1, 10, 135, 840, 3150, 7812, 13230, 15240, 11475, 5110, 1023

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

Previous definition: A007318 * a triangle by rows: for n > 0, n zeros followed by 2^n - 1.

Binomial transform of a diagonalized infinite lower triangular matrix with (1, 1, 3, 7, 15, ...) in the main diagonal and the rest zeros.

T(n,k) = |[1/(2^x)^k] 1 + (1-1/2^x)^n - (1-2/2^x)^n|. - Alois P. Heinz, Dec 10 2008

T(n,k) = binomial(n,k)*M(k) where M is Mersenne-like A255047. - Yuchun Ji, Feb 13 2019

Example

First few rows of the triangle:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  9,   7;
  1, 4, 18,  28,  15;
  1, 5, 30,  70,  75,  31;
  1, 6, 45, 140, 225, 186,  63;
  1, 7, 63, 245, 525, 651, 441, 127;
  ...

Maple

x:= 'x': T:= (n, k)-> `if` (k=0, 1, abs(coeff(expand((1-1/2^x)^n -(1-2/2^x)^n), 1/(2^x)^k))): seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Dec 10 2008
# Alternative:
T := (n, k) -> binomial(n, k)*(2^k - 1 + 0^k):
for n from 0 to 7 do seq(T(n, k), k=0..n) od;
# Or as a recursion:
p := proc(n, m) option remember; if n = 0 then max(1, m) else
    (m + x)*p(n - 1, m) - (m + 1)*p(n - 1, m + 1) fi end:
Trow := n -> seq((-1)^k * coeff(p(n, 0), x, n - k), k = 0..n):  # Peter Luschny, Jun 23 2023

Mathematica

max = 10; T1 = Table[Binomial[n, k], {n, 0, max}, {k, 0, max}]; T2 = Table[ If[n == k, 2^n-1, 0], {n, 0, max}, {k, 0, max}]; TT = T1.T2 ; T[_, 0]=1; T[n_, k_] := TT[[n+1, k+1]]; Table[T[n, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)

Crossrefs

Cf. A083313, A083323 (row sums), A255047 (main diagonal).

Sequence in context: A139633 A208330 A152440 * A135091 A171150 A111589

Adjacent sequences: A134316 A134317 A134318 * A134320 A134321 A134322

Keyword

nonn,tabl

Author

Gary W. Adamson, Oct 19 2007

Extensions

More terms from Alois P. Heinz, Dec 10 2008

New name using a formula of Yuchun Ji by Peter Luschny, Jun 23 2023

Status

approved