%I A093562
%S A093562 1,5,1,5,6,1,5,11,7,1,5,16,18,8,1,5,21,34,26,9,1,5,26,55,60,35,10,1,5,
%T A093562 31,81,115,95,45,11,1,5,36,112,196,210,140,56,12,1,5,41,148,308,406,350,
%U A093562 196,68,13,1,5,46,189,456,714,756,546,264,81,14,1,5,51,235,645,1170
%N A093562 (5,1) Pascal triangle.
%C A093562 This is the fifth member, d=5, in the family of triangles of figurate 
               numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, 
               A093560-1, for d=1,..,4.
%C A093562 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+4*z)/(1-(1+x)*z).
%C A093562 The SW-NE diagonals give A022095(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/
               2)), n>=1, with n=0 value 4. Observation by Paul Barry (pbarry(AT)wit.ie, 
               Apr 29 2004. Proof via recursion relations and comparison of inputs.
%C A093562 The array F(5;n,m) gives in the columns m>=1 the figurate numbers based 
               on A016861, including the heptagonal numbers A000566, (see the W. 
               Lang link).
%D A093562 Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek 
               Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D A093562 Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, 
               Berlin, 1993, ch.5, pp. 109-122.
%H A093562 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A093562.text">
               First 10 rows and array of figurate numbers </a>.
%F A093562 a(n, m)=F(5;n-m, m) for 0<= m <= n, else 0, with F(5;0, 0)=1, F(5;n, 
               0)=5 if n>=1 and F(5;n, m):=(5*n+m)*binomial(n+m-1, m-1)/m if m>=1.
%F A093562 G.f. column m (without leading zeros): (1+4*x)/(1-x)^(m+1), m>=0.
%F A093562 Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=5 if n>=1; a(n, m)= 
               a(n-1, m) + a(n-1, m-1).
%F A093562 T(n, k) = C(n, k) + 4*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%e A093562 [1]; [5,1]; [5,6,1]; [5,11,7,1]; ...
%Y A093562 Row sums: A007283(n-1), n>=1, 1 for n=0. A082505(n+1), alternating row 
               sums are 1 for n=0, 4 for n=2 and 0 else.
%Y A093562 Column sequences give for m=1..9: A016861, A000566 (heptagonal), A002413, 
               A002418, A027800, A051946, A050484, A052255, A055844.
%Y A093562 Cf. A093563 (d=6).
%Y A093562 Sequence in context: A087232 A151780 A054244 this_sequence A081774 A103193 
               A011093
%Y A093562 Adjacent sequences: A093559 A093560 A093561 this_sequence A093563 A093564 
               A093565
%K A093562 nonn,easy,tabl
%O A093562 0,2
%A A093562 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 22 2004