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A091913 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). 1
0, 1, 3, 2, 7, 9, 3, 15, 28, 18, 4, 31, 75, 70, 30, 5, 63, 186, 225, 140, 45, 6, 127, 441, 651, 525, 245, 63, 7, 255, 1016, 1764, 1736, 1050, 392, 84, 8, 511, 2295, 4572, 5292, 3906, 1890, 588, 108, 9, 1023, 5110, 11475, 15240, 13230, 7812, 3150, 840, 135, 10, 2047 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row lengths are 1,1,2,3,4,... = A028310. - M. F. Hasler, Jul 21 2012

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.

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

LINKS

Table of n, a(n) for n=0..56.

FORMULA

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]

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

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

EXAMPLE

Triangle begins

{0};

{1};

{3,2};

{7,9,3};

{15,28,18,4};

{31,75,70,30,5};

{63,186,225,140,45,6};

...

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

CROSSREFS

Cf. A007318, A038207.

Cf. A001787, A027641, A027642.

Sequence in context: A054170 A106167 A194473 * A212285 A192789 A026136

Adjacent sequences:  A091910 A091911 A091912 * A091914 A091915 A091916

KEYWORD

easy,nonn,tabf

AUTHOR

Ross La Haye, Mar 10 2004

EXTENSIONS

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

STATUS

approved

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Last modified November 18 14:30 EST 2017. Contains 294894 sequences.