The On-Line Encyclopedia of Integer Sequences (OEIS)

A134319 revision #19

A134319

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

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

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OFFSET

0,5

COMMENTS

Row sums = A083323: (1, 2, 6, 20, 66, 212, 666, ...).

LINKS

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

FORMULA

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, 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

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.

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

STATUS

proposed