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 A093564 (7,1) Pascal triangle. 13
 1, 7, 1, 7, 8, 1, 7, 15, 9, 1, 7, 22, 24, 10, 1, 7, 29, 46, 34, 11, 1, 7, 36, 75, 80, 45, 12, 1, 7, 43, 111, 155, 125, 57, 13, 1, 7, 50, 154, 266, 280, 182, 70, 14, 1, 7, 57, 204, 420, 546, 462, 252, 84, 15, 1, 7, 64, 261, 624, 966, 1008, 714, 336, 99, 16, 1, 7, 71, 325, 885 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The array F(7;n,m) gives in the columns m>=1 the figurate numbers based on A016993, including the 9-gonal numbers A001106, (see the W. Lang link). This is the seventh member, d=7, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-3, for d=1..6. 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+6*z)/(1-(1+x)*z). The SW-NE diagonals give A022097(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 6. 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(7;n-m, m) for 0<= m <= n, otherwise 0, with F(7;0, 0)=1, F(7;n, 0)=7 if n>=1 and F(7;n, m):=(7*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)=7 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1). G.f. column m (without leading zeros): (1+6*x)/(1-x)^(m+1), m>=0. T(n, k) = C(n, k) + 6*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)*(7 + 15*x + 9*x^2/2! + x^3/3!) = 7 + 22*x + 46*x^2/2! + 80*x^3/3! + 125*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];   [7,  1];   [7,  8,  1];   [7, 15,  9,  1];   ... MAPLE N:= 20: # to get the first N rows T:=Matrix(N, N): T[1, 1]:= 1: for m from 2 to N do T[m, 1]:= 7: T[m, 2..m]:= T[m-1, 1..m-1] + T[m-1, 2..m]; od: for m from 1 to N do convert(T[m, 1..m], list) od; # Robert Israel, Dec 28 2014 PROG (Haskell) a093564 n k = a093564_tabl !! n !! k a093564_row n = a093564_tabl !! n a093564_tabl = [1] : iterate                (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [7, 1] -- Reinhard Zumkeller, Sep 01 2014 CROSSREFS Row sums: A000079(n+2), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 6 for n=2 and 0 otherwise. The column sequences give for m=1..9: A016993, A001106 (9-gonal), A007584, A051740, A051877, A050403, A027818, A034266, A055994. Cf. A093565 (d=8). Sequence in context: A210708 A210529 A151785 * A081776 A256255 A131115 Adjacent sequences:  A093561 A093562 A093563 * A093565 A093566 A093567 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Apr 22 2004 STATUS approved

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Last modified March 26 20:47 EDT 2019. Contains 321535 sequences. (Running on oeis4.)