%I A093561
%S A093561 1,4,1,4,5,1,4,9,6,1,4,13,15,7,1,4,17,28,22,8,1,4,21,45,50,30,9,1,4,25,
%T A093561 66,95,80,39,10,1,4,29,91,161,175,119,49,11,1,4,33,120,252,336,294,168,
%U A093561 60,12,1,4,37,153,372,588,630,462,228,72,13,1,4,41,190,525,960,1218
%N A093561 (4,1) Pascal triangle.
%C A093561 The array F(4;n,m) gives in the columns m>=1 the figurate numbers based 
               on A016813, including the hexangonal numbers A000384, (see the W. 
               Lang link).
%C A093561 This is the fourth member, d=4, in the family of triangles of figurate 
               numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653 
               and A093560, for d=1,..,3.
%C A093561 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+3*z)/(1-(1+x)*z).
%C A093561 The SW-NE diagonals give A000285(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/
               2)), n>=1, with n=0 value 3. Observation by Paul Barry (pbarry(AT)wit.ie, 
               Apr 29 2004. Proof via recursion relations and comparison of inputs.
%D A093561 Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek 
               Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
%D A093561 Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, 
               Berlin, 1993, ch.5, pp. 109-122.
%H A093561 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A093561.text">
               First 10 rows and array of figurate numbers </a>.
%F A093561 a(n, m)=F(4;n-m, m) for 0<= m <= n, else 0, with F(4;0, 0)=1, F(4;n, 
               0)=4 if n>=1 and F(4;n, m):=(4*n+m)*binomial(n+m-1, m-1)/m if m>=1.
%F A093561 Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=4 if n>=1; a(n, m)= 
               a(n-1, m) + a(n-1, m-1).
%F A093561 G.f. row m (without leading zeros): (1+3*x)/(1-x)^(m+1), m>=0.
%F A093561 T(n, k) = C(n, k) + 3*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%e A093561 [1]; [4,1]; [4,5,1]; [4,9,6,1]; ...
%Y A093561 Row sums: A020714(n-1), n>=1, 1 for n=0, alternating row sums are 1 for 
               n=0, 3 for n=2 and 0 else.
%Y A093561 Columns m=1..9: A016813, A000384 (hexagonal), A002412, A002417, A034263, 
               A051947, A050483, A052181, A055843.
%Y A093561 Cf. A093562 (d=5).
%Y A093561 Sequence in context: A035646 A144034 A151783 this_sequence A081773 A167431 
               A110361
%Y A093561 Adjacent sequences: A093558 A093559 A093560 this_sequence A093562 A093563 
               A093564
%K A093561 nonn,easy,tabl
%O A093561 0,2
%A A093561 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 22 2004