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 A093561 (4,1) Pascal triangle. 15
 1, 4, 1, 4, 5, 1, 4, 9, 6, 1, 4, 13, 15, 7, 1, 4, 17, 28, 22, 8, 1, 4, 21, 45, 50, 30, 9, 1, 4, 25, 66, 95, 80, 39, 10, 1, 4, 29, 91, 161, 175, 119, 49, 11, 1, 4, 33, 120, 252, 336, 294, 168, 60, 12, 1, 4, 37, 153, 372, 588, 630, 462, 228, 72, 13, 1, 4, 41, 190, 525, 960, 1218 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The array F(4;n,m) gives in the columns m >= 1 the figurate numbers based on A016813, including the hexagonal numbers A000384 (see the W. Lang link). This is the fourth member, d=4, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653 and A093560, for d=1..3. 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+3*z)/(1-(1+x)*z). The SW-NE diagonals give A000285(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 3. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. 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(4;n-m, m) for 0<= m <= n, otherwise 0, with F(4;0, 0)=1, F(4;n, 0)=4 if n>=1 and F(4;n, m) = (4*n+m)*binomial(n+m-1, m-1)/m if m>=1. Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=4 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1). G.f. row m (without leading zeros): (1+3*x)/(1-x)^(m+1), m>=0. T(n, k) = C(n, k) + 3*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)*(4 + 9*x + 6*x^2/2! + x^3/3!) = 4 + 13*x + 28*x^2/2! + 50*x^3/3! + 80*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];   [4, 1];   [4, 5, 1];   [4, 9, 6, 1];   ... PROG (Haskell) a093561 n k = a093561_tabl !! n !! k a093561_row n = a093561_tabl !! n a093561_tabl = [1] : iterate                (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [4, 1] -- Reinhard Zumkeller, Aug 31 2014 CROSSREFS Cf. Row sums: A020714(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 3 for n=2 and 0 otherwise. Columns m=1..9: A016813, A000384 (hexagonal), A002412, A002417, A034263, A051947, A050483, A052181, A055843. Cf. A007318, A093562 (d=5), A228196, A228576. Sequence in context: A151783 A258853 A275791 * A286327 A081773 A302151 Adjacent sequences:  A093558 A093559 A093560 * A093562 A093563 A093564 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Apr 22 2004 STATUS approved

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Last modified May 27 03:48 EDT 2018. Contains 304690 sequences. (Running on oeis4.)