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A093562
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(5,1) Pascal triangle.
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13
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1, 5, 1, 5, 6, 1, 5, 11, 7, 1, 5, 16, 18, 8, 1, 5, 21, 34, 26, 9, 1, 5, 26, 55, 60, 35, 10, 1, 5, 31, 81, 115, 95, 45, 11, 1, 5, 36, 112, 196, 210, 140, 56, 12, 1, 5, 41, 148, 308, 406, 350, 196, 68, 13, 1, 5, 46, 189, 456, 714, 756, 546, 264, 81, 14, 1, 5, 51, 235, 645, 1170
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OFFSET
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0,2
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COMMENTS
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This is the fifth member, d=5, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-1, for d=1,..,4.
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(a(n,m)*x^m,m=0..n) is G(z,x)=(1+4*z)/(1-(1+x)*z).
The SW-NE diagonals give A022095(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with n=0 value 4. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
The array F(5;n,m) gives in the columns m>=1 the figurate numbers based on A016861, including the heptagonal numbers A000566, (see the W. Lang link).
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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..70.
W. Lang, First 10 rows and array of figurate numbers .
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FORMULA
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a(n, m)=F(5;n-m, m) for 0<= m <= n, else 0, with F(5;0, 0)=1, F(5;n, 0)=5 if n>=1 and F(5;n, m):=(5*n+m)*binomial(n+m-1, m-1)/m if m>=1.
G.f. column m (without leading zeros): (1+4*x)/(1-x)^(m+1), m>=0.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=5 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
T(n, k) = C(n, k) + 4*C(n-1, k) . - Philippe DELEHAM, Aug 28 2005
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EXAMPLE
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[1]; [5,1]; [5,6,1]; [5,11,7,1]; ...
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CROSSREFS
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Row sums: A007283(n-1), n>=1, 1 for n=0. A082505(n+1), alternating row sums are 1 for n=0, 4 for n=2 and 0 else.
Column sequences give for m=1..9: A016861, A000566 (heptagonal), A002413, A002418, A027800, A051946, A050484, A052255, A055844.
Cf. A093563 (d=6).
Sequence in context: A087232 A151780 A054244 * A081774 A103193 A011093
Adjacent sequences: A093559 A093560 A093561 * A093563 A093564 A093565
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang, Apr 22 2004
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STATUS
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approved
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