A093644 (9,1) Pascal triangle.

1, 9, 1, 9, 10, 1, 9, 19, 11, 1, 9, 28, 30, 12, 1, 9, 37, 58, 42, 13, 1, 9, 46, 95, 100, 55, 14, 1, 9, 55, 141, 195, 155, 69, 15, 1, 9, 64, 196, 336, 350, 224, 84, 16, 1, 9, 73, 260, 532, 686, 574, 308, 100, 17, 1, 9, 82, 333, 792, 1218, 1260, 882, 408, 117, 18, 1, 9, 91, 415

Offset: 0

Author

Wolfdieter Lang, Apr 22 2004

Keywords

• nonn
• easy
• tabl

Comments

The array F(9;n,m) gives in the columns m>=1 the figurate numbers based on A017173, including the 11-gonal numbers A051682 (see the W. Lang link).

This is the ninth member, d=9, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-5, for d=1..8.

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+8*z)/(1-(1+x)*z).

The SW-NE diagonals give A022099(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 8. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Triangle T(n,k), read by rows, given by (9,-8,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2011

Examples

			Triangle begins
  [1];
  [9,  1];
  [9, 10,  1];
  [9, 19, 11,  1];
  ...
		

References

• Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
• Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• Wolfdieter Lang, First 10 rows and array of figurate numbers .

Crossrefs

Row sums: A020714(n-1), n >= 1, 1 for n=0, alternating row sums are 1 for n=0, 8 for n=2 and 0 otherwise.

The column sequences give for m=1..9: A017173, A051682 (11-gonal), A007586, A051798, A051879, A050405, A052206, A056117, A056003.

Cf. A093645 (d=10).

Programs

Haskell

a093644 n k = a093644_tabl !! n !! k
a093644_row n = a093644_tabl !! n
a093644_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [9, 1]
-- Reinhard Zumkeller, Aug 31 2014

Mathematica

Join[{1},Table[Binomial[n,k]+8Binomial[n-1,k],{n,20},{k,0,n}]//Flatten] (* Harvey P. Dale, Aug 17 2024 *)

Formula

a(n, m) = F(9;n-m, m) for 0 <= m <= n, otherwise 0, with F(9;0, 0)=1, F(9;n, 0)=9 if n >= 1 and F(9;n, m):=(9*n+m)*binomial(n+m-1, m-1)/m if m >= 1.

Recursion: a(n, m)=0 if m > n, a(0, 0)= 1; a(n, 0)=9 if n >= 1; a(n, m) = a(n-1, m) + a(n-1, m-1).

G.f. column m (without leading zeros): (1+8*x)/(1-x)^(m+1), m >= 0.

T(n, k) = C(n, k) + 8*C(n-1, k). - Philippe Deléham, Aug 28 2005

Row n: Expansion of (9+x)*(1+x)^(n-1), n > 0. - Philippe Deléham, Oct 10 2011

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(9 + 19*x + 11*x^2/2! + x^3/3!) = 9 + 28*x + 58*x^2/2! + 100*x^3/3! + 155*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

G.f.: (-1-8*x)/(-1+x+x*y). - R. J. Mathar, Aug 11 2015