Search: id:A093645
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%I A093645
%S A093645 1,10,1,10,11,1,10,21,12,1,10,31,33,13,1,10,41,64,46,14,1,10,51,105,110,
%T A093645 60,15,1,10,61,156,215,170,75,16,1,10,71,217,371,385,245,91,17,1,10,81,
%U A093645 288,588,756,630,336,108,18,1,10,91,369,876,1344,1386,966,444,126,19,1
%N A093645 (10,1) Pascal triangle.
%C A093645 The array F(10;n,m) gives in the columns m>=1 the figurate numbers based 
               on A017281, including the 12-gonal numbers A051624, (see the W. Lang 
               link).
%C A093645 This is the tenth member, d=10, in the family of triangles of figurate 
               numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, 
               A093560-5 and A093644 for d=1,..,9.
%C A093645 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+9*z)/(1-(1+x)*z).
%C A093645 The SW-NE diagonals give A022100(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/
               2)), n>=1, with n=0 value 9. Observation by Paul Barry (pbarry(AT)wit.ie, 
               Apr 29 2004. Proof via recursion relations and comparison of inputs.
%D A093645 Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek 
               Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D A093645 Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, 
               Berlin, 1993, ch.5, pp. 109-122.
%H A093645 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A093645.text">
               First 10 rows and array of figurate numbers </a>.
%F A093645 a(n, m)=F(10;n-m, m) for 0<= m <= n, else 0, with F(10;0, 0)=1, F(10;
               n, 0)=10 if n>=1 and F(10;n, m):=(10*n+m)*binomial(n+m-1, m-1)/m 
               if m>=1.
%F A093645 Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=10 if n>=1; a(n, m)= 
               a(n-1, m) + a(n-1, m-1).
%F A093645 G.f. column m (without leading zeros): (1+9*x)/(1-x)^(m+1), m>=0.
%F A093645 T(n, k) = C(n, k) + 9*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%e A093645 [1]; [10,1]; [10,11,1]; [10,21,12,1]; ...
%Y A093645 Row sums: 1 for n=0 and A005015(n-1), n>=1, alternating row sums are 
               1 for n=0, 9 for n=2 and 0 else.
%Y A093645 The column sequences give for m=1..9: A017281, A051624 (12-gonal), A007587, 
               A051799, A051880, A050406, A052254, A056125, A093646.
%Y A093645 Sequence in context: A164915 A010691 A143970 this_sequence A130858 A098760 
               A107408
%Y A093645 Adjacent sequences: A093642 A093643 A093644 this_sequence A093646 A093647 
               A093648
%K A093645 nonn,easy,tabl
%O A093645 0,2
%A A093645 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 22 2004

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