A091913 - OEIS

%I #28 Feb 06 2022 06:52:07

%S 0,1,3,2,7,9,3,15,28,18,4,31,75,70,30,5,63,186,225,140,45,6,127,441,

%T 651,525,245,63,7,255,1016,1764,1736,1050,392,84,8,511,2295,4572,5292,

%U 3906,1890,588,108,9,1023,5110,11475,15240,13230,7812,3150,840,135,10,2047

%N Triangle read by rows: a(n,k) = C(n,k)*(2^(n-k) - 1) for k<n, a(n,k) = 0 for k >= n, where k=0..max(n-1,0).

%C Row lengths are 1,1,2,3,4,... = A028310. - _M. F. Hasler_, Jul 21 2012

%C Rows: Sum of the n-th row = A001047(n); Sum of the n-th row excluding column 0 = A028243(n+1). Columns: a(n,0) = A000225(n); a(n,1) = A058877(n). Diagonals: a(n,n-2) = A045943(n-1). Also note that the sums of the antidiagonals = A006684.

%C As an infinite lower triangular matrix * the Bernoulli numbers as a vector (Cf. A027641) = the natural numbers: [1, 2, 3, ...]. The same matrix * the Bernoulli number version starting [1, 1/2, 1/6, ...] = A001787: (1, 4, 12, 32, ...). - _Gary W. Adamson_, Mar 13 2012

%F For k>=n, a(n, k) = 0; for k < n, a(n, k) = C(n, k) * (2^(n-k) - 1) = Sum [C(n,k) * C(n-k, m), {m=1 to n-k}]. [Formula corrected Aug 22 2006]

%F The triangle (1; 3,2; 7,9,3; ...) = A007318^2 - A007318, then delete the right border of zeros. - _Gary W. Adamson_, Nov 16 2007

%F O.g.f.: 1/( (1 - (1 + x)*t)*(1 - (2 + x)*t) ) = 1 + (3 + 2*x)*t + (7 + 9*x + 3*x^2)*t^2 + .... - _Peter Bala_, Jul 16 2013

%e Triangle begins

%e    0;

%e    1;

%e    3,   2;

%e    7,   9,   3;

%e   15,  28,  18,   4;

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

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

%e   ...

%e a(5,3) = 30 because C(5,3) = 10, 2^(5 - 3) - 1 = 3 and 10 * 3 = 30.

%Y Cf. A007318, A038207.

%Y Cf. A001787, A027641, A027642.

%K easy,nonn,tabf

%O 0,3

%A _Ross La Haye_, Mar 10 2004

%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004