A134319 - OEIS

%I #25 Jun 23 2023 10:31:44

%S 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,

%T 63,1,7,63,245,525,651,441,127,1,8,84,392,1050,1736,1764,1016,255,1,9,

%U 108,588,1890,3906,5292,4572,2295,511,1,10,135,840,3150,7812,13230,15240,11475,5110,1023

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

%H Alois P. Heinz, <a href="/A134319/b134319.txt">Rows n = 0..140, flattened</a>

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

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

%F T(n,k) = |[1/(2^x)^k] 1 + (1-1/2^x)^n - (1-2/2^x)^n|. - _Alois P. Heinz_, Dec 10 2008

%F T(n,k) = binomial(n,k)*M(k) where M is Mersenne-like A255047. - _Yuchun Ji_, Feb 13 2019

%e First few rows of the triangle:

%e   1;

%e   1, 1;

%e   1, 2,  3;

%e   1, 3,  9,   7;

%e   1, 4, 18,  28,  15;

%e   1, 5, 30,  70,  75,  31;

%e   1, 6, 45, 140, 225, 186,  63;

%e   1, 7, 63, 245, 525, 651, 441, 127;

%e   ...

%p 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

%p # Alternative:

%p T := (n, k) -> binomial(n, k)*(2^k - 1 + 0^k):

%p for n from 0 to 7 do seq(T(n, k), k=0..n) od;

%p # Or as a recursion:

%p p := proc(n, m) option remember; if n = 0 then max(1, m) else

%p     (m + x)*p(n - 1, m) - (m + 1)*p(n - 1, m + 1) fi end:

%p Trow := n -> seq((-1)^k * coeff(p(n, 0), x, n - k), k = 0..n):  # _Peter Luschny_, Jun 23 2023

%t 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 *)

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

%K nonn,tabl

%O 0,5

%A _Gary W. Adamson_, Oct 19 2007

%E More terms from _Alois P. Heinz_, Dec 10 2008

%E New name using a formula of _Yuchun Ji_ by _Peter Luschny_, Jun 23 2023