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Search: id:A093645
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| A093645 |
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(10,1) Pascal triangle. |
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27
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| 1, 10, 1, 10, 11, 1, 10, 21, 12, 1, 10, 31, 33, 13, 1, 10, 41, 64, 46, 14, 1, 10, 51, 105, 110, 60, 15, 1, 10, 61, 156, 215, 170, 75, 16, 1, 10, 71, 217, 371, 385, 245, 91, 17, 1, 10, 81, 288, 588, 756, 630, 336, 108, 18, 1, 10, 91, 369, 876, 1344, 1386, 966, 444, 126, 19, 1
(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(10;n,m) gives in the columns m>=1 the figurate numbers based on A017281, including the 12-gonal numbers A051624, (see the W. Lang link).
This is the tenth member, d=10, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-5 and A093644 for d=1,..,9.
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+9*z)/(1-(1+x)*z).
The SW-NE diagonals give A022100(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with n=0 value 9. 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(10;n-m, m) for 0<= m <= n, else 0, with F(10;0, 0)=1, F(10;n, 0)=10 if n>=1 and F(10;n, m):=(10*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)=10 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+9*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 9*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 28 2005
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EXAMPLE
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[1]; [10,1]; [10,11,1]; [10,21,12,1]; ...
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CROSSREFS
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Row sums: 1 for n=0 and A005015(n-1), n>=1, alternating row sums are 1 for n=0, 9 for n=2 and 0 else.
The column sequences give for m=1..9: A017281, A051624 (12-gonal), A007587, A051799, A051880, A050406, A052254, A056125, A093646.
Sequence in context: A164915 A010691 A143970 this_sequence A130858 A098760 A107408
Adjacent sequences: A093642 A093643 A093644 this_sequence A093646 A093647 A093648
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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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