Search: id:A093564
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%I A093564
%S A093564 1,7,1,7,8,1,7,15,9,1,7,22,24,10,1,7,29,46,34,11,1,7,36,75,80,45,12,1,
               7,
%T A093564 43,111,155,125,57,13,1,7,50,154,266,280,182,70,14,1,7,57,204,420,546,
%U A093564 462,252,84,15,1,7,64,261,624,966,1008,714,336,99,16,1,7,71,325,885
%N A093564 (7,1) Pascal triangle.
%C A093564 The array F(7;n,m) gives in the columns m>=1 the figurate numbers based 
               on A016993, including the 9-gonal numbers A001106, (see the W. Lang 
               link).
%C A093564 This is the seventh member, d=7, in the family of triangles of figurate 
               numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, 
               A093560-3, for d=1,..,6.
%C A093564 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+6*z)/(1-(1+x)*z).
%C A093564 The SW-NE diagonals give A022097(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/
               2)), n>=1, with n=0 value 6. Observation by Paul Barry (pbarry(AT)wit.ie, 
               Apr 29 2004. Proof via recursion relations and comparison of inputs.
%D A093564 Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek 
               Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D A093564 Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, 
               Berlin, 1993, ch.5, pp. 109-122.
%H A093564 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A093564.text">
               First 10 rows and array of figurate numbers </a>.
%F A093564 a(n, m)=F(7;n-m, m) for 0<= m <= n, else 0, with F(7;0, 0)=1, F(7;n, 
               0)=7 if n>=1 and F(7;n, m):=(7*n+m)*binomial(n+m-1, m-1)/m if m>=1.
%F A093564 Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=7 if n>=1; a(n, m)= 
               a(n-1, m) + a(n-1, m-1).
%F A093564 G.f. column m (without leading zeros): (1+6*x)/(1-x)^(m+1), m>=0.
%F A093564 T(n, k) = C(n, k) + 6*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%e A093564 [1]; [7,1]; [7,8,1]; [7,15,9,1]; ...
%Y A093564 Row sums: A000079(n+2), n>=1, 1 for n=0, alternating row sums are 1 for 
               n=0, 6 for n=2 and 0 else.
%Y A093564 The column sequences give for m=1..9: A016993, A001106 (9-gonal), A007584, 
               A051740, A051877, A050403, A027818, A034266, A055994.
%Y A093564 Cf. A093565 (d=8).
%Y A093564 Sequence in context: A033953 A010772 A151785 this_sequence A081776 A131115 
               A151778
%Y A093564 Adjacent sequences: A093561 A093562 A093563 this_sequence A093565 A093566 
               A093567
%K A093564 nonn,easy,tabl
%O A093564 0,2
%A A093564 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 22 2004

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