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 A093562 (5,1) Pascal triangle. 15
 1, 5, 1, 5, 6, 1, 5, 11, 7, 1, 5, 16, 18, 8, 1, 5, 21, 34, 26, 9, 1, 5, 26, 55, 60, 35, 10, 1, 5, 31, 81, 115, 95, 45, 11, 1, 5, 36, 112, 196, 210, 140, 56, 12, 1, 5, 41, 148, 308, 406, 350, 196, 68, 13, 1, 5, 46, 189, 456, 714, 756, 546, 264, 81, 14, 1, 5, 51, 235, 645, 1170 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is the fifth member, d=5, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-1, for d=1..4. This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). 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)=(1+4*z)/(1-(1+x)*z). The SW-NE diagonals give A022095(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 4. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. The array F(5;n,m) gives in the columns m >= 1 the figurate numbers based on A016861, including the heptagonal numbers A000566 (see the W. Lang link). For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013 The n-th row polynomial is (4 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018 REFERENCES Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen. Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122. LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened W. Lang, First 10 rows and array of figurate numbers . FORMULA a(n, m) = F(5;n-m, m) for 0<= m <= n, otherwise 0, with F(5;0, 0)=1, F(5;n, 0)=5 if n>=1 and F(5;n, m):=(5*n+m)*binomial(n+m-1, m-1)/m if m>=1. G.f. column m (without leading zeros): (1+4*x)/(1-x)^(m+1), m>=0. Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=5 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1). T(n, k) = C(n, k) + 4*C(n-1, k). - Philippe Deléham, Aug 28 2005 exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(5 + 11*x + 7*x^2/2! + x^3/3!) = 5 + 16*x + 34*x^2/2! + 60*x^3/3! + 95*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014 EXAMPLE Triangle begins   [1];   [5,  1];   [5,  6,  1];   [5, 11,  7,  1];   ... PROG (Haskell) a093562 n k = a093562_tabl !! n !! k a093562_row n = a093562_tabl !! n a093562_tabl = [1] : iterate                (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [5, 1] -- Reinhard Zumkeller, Aug 31 2014 CROSSREFS Cf. Row sums: A007283(n-1), n>=1, 1 for n=0. A082505(n+1), alternating row sums are 1 for n=0, 4 for n=2 and 0 else. Column sequences give for m=1..9: A016861, A000566 (heptagonal), A002413, A002418, A027800, A051946, A050484, A052255, A055844. A007318, A093563 (d=6), A228196, A228576. Sequence in context: A087232 A151780 A054244 * A081774 A103193 A011093 Adjacent sequences:  A093559 A093560 A093561 * A093563 A093564 A093565 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Apr 22 2004 STATUS approved

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Last modified October 14 07:31 EDT 2019. Contains 327995 sequences. (Running on oeis4.)