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A093563 (6,1)-Pascal triangle. 16
1, 6, 1, 6, 7, 1, 6, 13, 8, 1, 6, 19, 21, 9, 1, 6, 25, 40, 30, 10, 1, 6, 31, 65, 70, 40, 11, 1, 6, 37, 96, 135, 110, 51, 12, 1, 6, 43, 133, 231, 245, 161, 63, 13, 1, 6, 49, 176, 364, 476, 406, 224, 76, 14, 1, 6, 55, 225, 540, 840, 882, 630, 300, 90, 15, 1, 6, 61, 280, 765, 1380 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The array F(6;n,m) gives in the columns m >= 1 the figurate numbers based on A016921, including the octagonal numbers A000567, (see the W. Lang link).

This is the sixth member, d=6, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-2, for d=1..5.

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+5*z)/(1-(1+x)*z).

The SW-NE diagonals give A022096(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 5. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

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

FORMULA

a(n, m)=F(6;n-m, m) for 0<= m <= n, otherwise 0, with F(6;0, 0)=1, F(6;n, 0)=6 if n>=1 and F(6;n, m):= (6*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)=6 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).

G.f. column m (without leading zeros): (1+5*x)/(1-x)^(m+1), m>=0.

T(n, k) = C(n, k) + 5*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)*(6 + 13*x + 8*x^2/2! + x^3/3!) = 6 + 19*x + 40*x^2/2! + 70*x^3/3! + 110*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;

  6,  1;

  6,  7,  1;

  6, 13,  8,  1;

  6, 19, 21,  9,  1;

  6, 25, 40, 30, 10,  1;

  ...

MATHEMATICA

lim = 11; s = Series[(1 + 5*x)/(1 - x)^(m + 1), {x, 0, lim}]; t = Table[ CoefficientList[s, x], {m, 0, lim}]; Flatten[ Table[t[[j - k + 1, k]], {j, lim + 1}, {k, j, 1, -1}]] (* Jean-François Alcover, Sep 16 2011, after g.f. *)

PROG

(Haskell)

a093563 n k = a093563_tabl !! n !! k

a093563_row n = a093563_tabl !! n

a093563_tabl = [1] : iterate

               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [6, 1]

-- Reinhard Zumkeller, Aug 31 2014

CROSSREFS

Row sums: A005009(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 5 for n=2 and 0 else.

Cf. A007318, A093564 (d=7), A228196, A228576.

The column sequences give for m=1..9: A016921, A000567 (octagonal), A002414, A002419, A051843, A027810, A034265, A054487, A055848.

Sequence in context: A292862 A070472 A151784 * A081775 A156163 A301817

Adjacent sequences:  A093560 A093561 A093562 * A093564 A093565 A093566

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Apr 22 2004

STATUS

approved

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Last modified August 22 03:39 EDT 2018. Contains 313964 sequences. (Running on oeis4.)