Search: id:A093560
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%I A093560
%S A093560 1,3,1,3,4,1,3,7,5,1,3,10,12,6,1,3,13,22,18,7,1,3,16,35,40,25,8,1,3,19,
%T A093560 51,75,65,33,9,1,3,22,70,126,140,98,42,10,1,3,25,92,196,266,238,140,52,
%U A093560 11,1,3,28,117,288,462,504,378,192,63,12,1,3,31,145,405,750,966,882,570
%N A093560 (3,1) Pascal triangle.
%C A093560 The array F(3;n,m) gives in the columns m>=1 the figurate numbers based 
               on A016777, including the pentagonal numbers A000326, (see the W. 
               Lang link).
%C A093560 This is the third member, d=3, in the family of triangles of figurate 
               numbers, called (d,1) Pascal triangles: A007318 (Pascal (d=1), A029653 
               (d=2).
%C A093560 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(a(n,m)*x^m,m=0..n) 
               is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(1+2*x)/(1-x), f(x)=1/(1-x), 
               hence G(z,x)=(1+2*z)/(1-(1+x)*z).
%C A093560 The SW-NE diagonals give the Lucas numbers A000032: L(n)= sum( a(n-1-k,
               k),k=0..ceiling((n-1)/2)), n>=1, with L(0)=2. Observation by Paul 
               Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations 
               and comparison of inputs.
%C A093560 Triangle T(n,k), read by rows, given by [3,-2,0,0,0,0,0,0,...] DELTA 
               [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 
               . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]
%D A093560 Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek 
               Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D A093560 Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, 
               Berlin, 1993, ch.5, pp. 109-122.
%H A093560 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A093560.text">
               First 10 rows and array of figurate numbers </a>.
%F A093560 a(n, m)=F(3;n-m, m) for 0<= m <= n, else 0, with F(3;0, 0)=1, F(3;n, 
               0)=3 if n>=1 and F(3;n, m):=(3*n+m)*binomial(n+m-1, m-1)/m if m>=1.
%F A093560 G.f. column m (without leading zeros): (1+2*x)/(1-x)^(m+1), m>=0.
%F A093560 Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=3 if n>=1; a(n, m)= 
               a(n-1, m) + a(n-1, m-1).
%F A093560 T(n, k) = C(n, k) + 2*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%F A093560 Equals M * A007318, where M = an infinite triangular matrix with all 
               1's in the main diagonal and all 2's in the subdiagonal. - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Dec 01 2007
%F A093560 Sum_{k, 0<=k<=n}T(n,k)=A151821(n+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 17 2009]
%e A093560 [1]; [3,1]; [3,4,1]; [3,7,5,1]; ...
%Y A093560 Column sequences for m=1..9: A016777, A000326 (pentagonal), A002411, 
               A001296, A051836, A051923, A050494, A053367, A053310.
%Y A093560 Cf. A029653 (2, 1) Pascal triangle.
%Y A093560 Cf. A093561(d=4).
%Y A093560 Sequence in context: A107638 A104765 A064884 this_sequence A131504 A008311 
               A081772
%Y A093560 Adjacent sequences: A093557 A093558 A093559 this_sequence A093561 A093562 
               A093563
%K A093560 nonn,tabl,easy
%O A093560 0,2
%A A093560 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 22 2004
%E A093560 Incorrect connection with A046055 deleted by N. J. A. Sloane, Jul 08 
               2009

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