Search: id:A093565
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%I A093565
%S A093565 1,8,1,8,9,1,8,17,10,1,8,25,27,11,1,8,33,52,38,12,1,8,41,85,90,50,13,1,
%T A093565 8,49,126,175,140,63,14,1,8,57,175,301,315,203,77,15,1,8,65,232,476,616,
%U A093565 518,280,92,16,1,8,73,297,708,1092,1134,798,372,108,17,1,8,81,370,1005
%N A093565 (8,1) Pascal triangle.
%C A093565 The array F(8;n,m) gives in the columns m>=1 the figurate numbers based 
               on A017077, including the decagonal numbers A001107,(see the W. Lang 
               link).
%C A093565 This is the eighth member, d=8, in the family of triangles of figurate 
               numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, 
               A093560-4, for d=1,..,7.
%C A093565 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+7*z)/(1-(1+x)*z).
%C A093565 The SW-NE diagonals give A022098(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/
               2)), n>=1, with n=0 value 7. Observation by Paul Barry (pbarry(AT)wit.ie, 
               Apr 29 2004. Proof via recursion relations and comparison of inputs.
%D A093565 Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek 
               Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D A093565 Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, 
               Berlin, 1993, ch.5, pp. 109-122.
%H A093565 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A093565.text">
               First 10 rows and array of figurate numbers </a>.
%F A093565 a(n, m)=F(8;n-m, m) for 0<= m <= n, else 0, with F(8;0, 0)=1, F(8;n, 
               0)=8 if n>=1 and F(8;n, m):=(8*n+m)*binomial(n+m-1, m-1)/m if m>=1.
%F A093565 Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=8 if n>=1; a(n, m)= 
               a(n-1, m) + a(n-1, m-1).
%F A093565 G.f. column m (without leading zeros): (1+7*x)/(1-x)^(m+1), m>=0.
%F A093565 T(n, k) = C(n, k) + 7*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%e A093565 [1]; [8,1]; [8,9,1]; [8,17,10,1]; ...
%Y A093565 Row sums: A005010(n-1), n>=1, 1 for n=0, alternating row sums are 1 for 
               n=0, 7 for n=2 and 0 else.
%Y A093565 The column sequences give for m=1..9: A017077, A001107 (decagonal), A007585, 
               A051797, A051878, A050404, A052226, A056001, A056122.
%Y A093565 Cf. A093644 (d=9).
%Y A093565 Sequence in context: A092618 A151786 A094770 this_sequence A081777 A098367 
               A141228
%Y A093565 Adjacent sequences: A093562 A093563 A093564 this_sequence A093566 A093567 
               A093568
%K A093565 nonn,easy,tabl
%O A093565 0,2
%A A093565 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 22 2004

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