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A093560
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(3,1) Pascal triangle.
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18
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1, 3, 1, 3, 4, 1, 3, 7, 5, 1, 3, 10, 12, 6, 1, 3, 13, 22, 18, 7, 1, 3, 16, 35, 40, 25, 8, 1, 3, 19, 51, 75, 65, 33, 9, 1, 3, 22, 70, 126, 140, 98, 42, 10, 1, 3, 25, 92, 196, 266, 238, 140, 52, 11, 1, 3, 28, 117, 288, 462, 504, 378, 192, 63, 12, 1, 3, 31, 145, 405, 750, 966, 882, 570
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OFFSET
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0,2
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COMMENTS
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The array F(3;n,m) gives in the columns m>=1 the figurate numbers based on A016777, including the pentagonal numbers A000326, (see the W. Lang link).
This is the third member, d=3, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal (d=1), A029653 (d=2).
This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column nr. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n) is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(1+2*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(1+2*z)/(1-(1+x)*z).
The SW-NE diagonals give the Lucas numbers A000032: L(n)= sum( a(n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with L(0)=2. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Triangle T(n,k), read by rows, given by [3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . [From Philippe DELEHAM, Sep 17 2009]
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REFERENCES
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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, Birkhaeuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.
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LINKS
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Table of n, a(n) for n=0..73.
W. Lang, First 10 rows and array of figurate numbers .
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FORMULA
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a(n, m)=F(3;n-m, m) for 0<= m <= n, else 0, with F(3;0, 0)=1, F(3;n, 0)=3 if n>=1 and F(3;n, m):=(3*n+m)*binomial(n+m-1, m-1)/m if m>=1.
G.f. column m (without leading zeros): (1+2*x)/(1-x)^(m+1), m>=0.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=3 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
T(n, k) = C(n, k) + 2*C(n-1, k) . - Philippe DELEHAM, Aug 28 2005
Equals M * A007318, where M = an infinite triangular matrix with all 1's in the main diagonal and all 2's in the subdiagonal. - Gary W. Adamson, Dec 01 2007
Sum_{k, 0<=k<=n}T(n,k)=A151821(n+1). [From Philippe DELEHAM, Sep 17 2009]
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EXAMPLE
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[1]; [3,1]; [3,4,1]; [3,7,5,1]; ...
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CROSSREFS
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Column sequences for m=1..9: A016777, A000326 (pentagonal), A002411, A001296, A051836, A051923, A050494, A053367, A053310.
Cf. A029653 (2, 1) Pascal triangle.
Cf. A093561(d=4).
Sequence in context: A107638 A104765 A064884 * A173934 A131504 A008311
Adjacent sequences: A093557 A093558 A093559 * A093561 A093562 A093563
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Wolfdieter Lang, Apr 22 2004
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EXTENSIONS
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Incorrect connection with A046055 deleted by N. J. A. Sloane, Jul 08 2009
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STATUS
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approved
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