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%I #39 Aug 28 2019 16:09:30
%S 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,
%T 31,81,115,95,45,11,1,5,36,112,196,210,140,56,12,1,5,41,148,308,406,
%U 350,196,68,13,1,5,46,189,456,714,756,546,264,81,14,1,5,51,235,645,1170
%N (5,1) Pascal triangle.
%C 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.
%C 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).
%C 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.
%C 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).
%C For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013
%C 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
%D Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.
%H Reinhard Zumkeller, <a href="/A093562/b093562.txt">Rows n = 0..125 of triangle, flattened</a>
%H P. Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%H W. Lang, <a href="/A093562/a093562.txt">First 10 rows and array of figurate numbers </a>.
%F 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.
%F G.f. column m (without leading zeros): (1+4*x)/(1-x)^(m+1), m>=0.
%F 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).
%F T(n, k) = C(n, k) + 4*C(n-1, k). - _Philippe Deléham_, Aug 28 2005
%F 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
%e Triangle begins
%e [1];
%e [5, 1];
%e [5, 6, 1];
%e [5, 11, 7, 1];
%e ...
%o (Haskell)
%o a093562 n k = a093562_tabl !! n !! k
%o a093562_row n = a093562_tabl !! n
%o a093562_tabl = [1] : iterate
%o (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [5, 1]
%o -- _Reinhard Zumkeller_, Aug 31 2014
%Y 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.
%Y Column sequences give for m=1..9: A016861, A000566 (heptagonal), A002413, A002418, A027800, A051946, A050484, A052255, A055844.
%Y A007318, A093563 (d=6), A228196, A228576.
%K nonn,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Apr 22 2004
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