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A093565 (8,1) Pascal triangle. 14
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, 1, 8, 49, 126, 175, 140, 63, 14, 1, 8, 57, 175, 301, 315, 203, 77, 15, 1, 8, 65, 232, 476, 616, 518, 280, 92, 16, 1, 8, 73, 297, 708, 1092, 1134, 798, 372, 108, 17, 1, 8, 81, 370, 1005 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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.

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).

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.

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

W. Lang, First 10 rows and array of figurate numbers .

FORMULA

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.

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).

G.f. column m (without leading zeros): (1+7*x)/(1-x)^(m+1), m>=0.

T(n, k) = C(n, k) + 7*C(n-1, k). - Philippe Deléham, Aug 28 2005

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

EXAMPLE

Triangle begins

  [1];

  [8,  1];

  [8,  9,  1];

  [8, 17, 10,  1];

  ...

PROG

(Haskell)

a093565 n k = a093565_tabl !! n !! k

a093565_row n = a093565_tabl !! n

a093565_tabl = [1] : iterate

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

-- Reinhard Zumkeller, Aug 31 2014

CROSSREFS

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.

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

Cf. A093644 (d=9).

Sequence in context: A092618 A151786 A094770 * A081777 A198988 A098367

Adjacent sequences:  A093562 A093563 A093564 * A093566 A093567 A093568

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Apr 22 2004

STATUS

approved

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Last modified October 21 04:12 EDT 2018. Contains 316405 sequences. (Running on oeis4.)