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Search: id:A093563
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| A093563 |
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(6,1)-Pascal triangle. |
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+0
14
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| 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; listen)
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OFFSET
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0,2
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COMMENT
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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(a(n,m)*x^m,m=0..n) is G(z,x)=(1+5*z)/(1-(1+x)*z).
The SW-NE diagonals give A022096(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with n=0 value 5. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs.
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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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W. Lang, First 10 rows and array of figurate numbers .
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FORMULA
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a(n, m)=F(6;n-m, m) for 0<= m <= n, else 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 DELEHAM (kolotoko(AT)wanadoo.fr), Aug 28 2005
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EXAMPLE
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[1]; [6,1]; [6,7,1]; [6,13,8,1]; ...
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CROSSREFS
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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. A093564 (d=7).
The column sequences give for m=1..9: A016921, A000567 (octogonal), A002414, A002419, A051843, A027810, A034265, A054487, A055848.
Sequence in context: A070514 A070472 A151784 this_sequence A081775 A156163 A011300
Adjacent sequences: A093560 A093561 A093562 this_sequence A093564 A093565 A093566
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Apr 22 2004
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