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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 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 ; [1,6]; [1,7,6]; [1,8,13,6]; [1,9,21,16,6]; ... MAPLE a(n, k):=piecewise(n=0, 6, 0=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.)