Search: id:A093563
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%I A093563
%S A093563 1,6,1,6,7,1,6,13,8,1,6,19,21,9,1,6,25,40,30,10,1,6,31,65,70,40,11,1,6,
%T A093563 37,96,135,110,51,12,1,6,43,133,231,245,161,63,13,1,6,49,176,364,476,
%U A093563 406,224,76,14,1,6,55,225,540,840,882,630,300,90,15,1,6,61,280,765,1380
%N A093563 (6,1)-Pascal triangle.
%C A093563 The array F(6;n,m) gives in the columns m>=1 the figurate numbers based 
               on A016921, including the octagonal numbers A000567, (see the W. 
               Lang link).
%C A093563 This is the sixth member, d=6, in the family of triangles of figurate 
               numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, 
               A093560-2, for d=1,..,5.
%C A093563 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(a(n,m)*x^m,m=0..n) is G(z,x)=(1+5*z)/(1-(1+x)*z).
%C A093563 The SW-NE diagonals give A022096(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/
               2)), n>=1, with n=0 value 5. Observation by Paul Barry (pbarry(AT)wit.ie, 
               Apr 29 2004. Proof via recursion relations and comparison of inputs.
%D A093563 Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek 
               Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D A093563 Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, 
               Berlin, 1993, ch.5, pp. 109-122.
%H A093563 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A093563.text">
               First 10 rows and array of figurate numbers </a>.
%F A093563 a(n, m)=F(6;n-m, m) for 0<= m <= n, else 0, with F(6;0, 0)=1, F(6;n, 
               0)=6 if n>=1 and F(6;n, m):= (6*n+m)*binomial(n+m-1, m-1)/m if m>
               =1
%F A093563 Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=6 if n>=1; a(n, m)= 
               a(n-1, m) + a(n-1, m-1).
%F A093563 G.f. column m (without leading zeros): (1+5*x)/(1-x)^(m+1), m>=0.
%F A093563 T(n, k) = C(n, k) + 5*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%e A093563 [1]; [6,1]; [6,7,1]; [6,13,8,1]; ...
%Y A093563 Row sums: A005009(n-1), n>=1, 1 for n=0, alternating row sums are 1 for 
               n=0, 5 for n=2 and 0 else.
%Y A093563 Cf. A093564 (d=7).
%Y A093563 The column sequences give for m=1..9: A016921, A000567 (octogonal), A002414, 
               A002419, A051843, A027810, A034265, A054487, A055848.
%Y A093563 Sequence in context: A070514 A070472 A151784 this_sequence A081775 A156163 
               A011300
%Y A093563 Adjacent sequences: A093560 A093561 A093562 this_sequence A093564 A093565 
               A093566
%K A093563 nonn,easy,tabl
%O A093563 0,2
%A A093563 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 22 2004

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