A093565 - OEIS

%I #26 Aug 28 2019 16:10:15

%S 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,

%T 1,8,49,126,175,140,63,14,1,8,57,175,301,315,203,77,15,1,8,65,232,476,

%U 616,518,280,92,16,1,8,73,297,708,1092,1134,798,372,108,17,1,8,81,370,1005

%N (8,1) Pascal triangle.

%C 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 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 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_{m=0..n} a(n,m)*x^m is G(z,x)=(1+7*z)/(1-(1+x)*z).

%C The SW-NE diagonals give A022098(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 7. Observation by _Paul Barry_, Apr 29 2004. Proof via recursion relations and comparison of inputs.

%D Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.

%D Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

%H Reinhard Zumkeller, <a href="/A093565/b093565.txt">Rows n = 0..125 of triangle, flattened</a>

%H W. Lang, <a href="/A093565/a093565.txt">First 10 rows and array of figurate numbers </a>.

%F a(n, m)=F(8;n-m, m) for 0<= m <= n, otherwise 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 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 G.f. column m (without leading zeros): (1+7*x)/(1-x)^(m+1), m>=0.

%F T(n, k) = C(n, k) + 7*C(n-1, k). - _Philippe Deléham_, Aug 28 2005

%F exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(8 + 17*x + 10*x^2/2! + x^3/3!) = 8 + 25*x + 52*x^2/2! + 90*x^3/3! + 140*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 22 2014

%e Triangle begins

%e   [1];

%e   [8,  1];

%e   [8,  9,  1];

%e   [8, 17, 10,  1];

%e   ...

%o (Haskell)

%o a093565 n k = a093565_tabl !! n !! k

%o a093565_row n = a093565_tabl !! n

%o a093565_tabl = [1] : iterate

%o                (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [8, 1]

%o -- _Reinhard Zumkeller_, Aug 31 2014

%Y 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 The column sequences give for m=1..9: A017077, A001107 (decagonal), A007585, A051797, A051878, A050404, A052226, A056001, A056122.

%Y Cf. A093644 (d=9).

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Apr 22 2004