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A096956 Pascal (1,6) triangle. 11
6, 1, 6, 1, 7, 6, 1, 8, 13, 6, 1, 9, 21, 19, 6, 1, 10, 30, 40, 25, 6, 1, 11, 40, 70, 65, 31, 6, 1, 12, 51, 110, 135, 96, 37, 6, 1, 13, 63, 161, 245, 231, 133, 43, 6, 1, 14, 76, 224, 406, 476, 364, 176, 49, 6, 1, 15, 90, 300, 630, 882, 840, 540, 225, 55, 6, 1, 16, 105, 390, 930 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Except for the first row this is the row reversed (6,1)-Pascal triangle A093563.

This is the sixth member, q=6, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1), A029635 (q=2) (but with a(0,0)=2, not 1), A095660, A095666, A096940.

This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column nr. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(6-5*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(6-5*z)/(1-(1+x)*z).

The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k) = A022097(n-2), n >= 2, with n=1 value 6. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

LINKS

Table of n, a(n) for n=0..70.

W. Lang, First 10 rows.

FORMULA

Recursion: a(n,m)=0 if m > n, a(0,0) = 6; a(n,0) = 1 if n >= 1; a(n,m) = a(n-1, m) + a(n-1, m-1).

G.f. column m (without leading zeros): (6-5*x)/(1-x)^(m+1), m >= 0.

a(n,k) = (1+5*k/n)*binomial(n,k)), for n > 0. - Mircea Merca, Apr 08 2012

EXAMPLE

[6]; [1,6]; [1,7,6]; [1,8,13,6]; [1,9,21,16,6]; ...

MAPLE

a(n, k):=piecewise(n=0, 6, 0<n, (1+5*k/n)*binomial(n, k)) # Mircea Merca, Apr 08 2012

CROSSREFS

Row sums: A005009(n-1), n>=1, 6 if n=0; g.f.: (6-5*x)/(1-2*x). Alternating row sums are [6, -5, followed by 0's].

Column sequences (without leading zeros) give for m=1..9, with n >= 0: A000027(n+6), A056115, A096957-9, A097297-A097300.

Sequence in context: A176355 A109918 A263494 * A326126 A176398 A078300

Adjacent sequences:  A096953 A096954 A096955 * A096957 A096958 A096959

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Aug 13 2004

STATUS

approved

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Last modified February 25 11:37 EST 2020. Contains 332233 sequences. (Running on oeis4.)