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A111125
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Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
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10
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1, 3, 1, 5, 5, 1, 7, 14, 7, 1, 9, 30, 27, 9, 1, 11, 55, 77, 44, 11, 1, 13, 91, 182, 156, 65, 13, 1, 15, 140, 378, 450, 275, 90, 15, 1, 17, 204, 714, 1122, 935, 442, 119, 17, 1, 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1, 21, 385, 2079, 5148, 7007, 5733, 2940, 952
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Riordan array ((1+x)/(1-x)^2,x/(1-x)^2). Row sums are A002878. Diagonal sums are A003945. Inverse is A113187. An interesting factorization is (1/(1-x),x/(1-x))(1+2x,x(1+x)). - Paul Barry (pbarry(AT)wit.ie), Oct 17 2005
Central coefficients of rows with odd numbers of term are A052227.
From Wolfdieter Lang, Jun 26, 2011: (Start)
T(k,s) appears as T_s(k) in the Knuth reference, p. 285.
This triangle is related to triangle A156308(n,m), appearing in this reference as U_m(n) on p. 285, by T(k,s)-T(k-1,s) = A156308(k,s), k>=s>=1 (identity on p.286).
T(k,s) = A156308(k+1,s+1) - A156308(k,s+1), k>=s>=0 (identity on p. 286).
(End)
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REFERENCES
| K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370.
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LINKS
| D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.
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FORMULA
| T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
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EXAMPLE
| Triangle begins:
1
3 1
5 5 1
7 14 7 1
9 30 27 9 1
11 55 77 44 11 1
13 91 182 156 65 13 1
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CROSSREFS
| Mirror image of A082985, which see for further references, etc.
Also closely related to triangles in A098599 and A100218.
A052227.
Sequence in context: A131768 A084533 A082985 * A072919 A104489 A067285
Adjacent sequences: A111122 A111123 A111124 * A111126 A111127 A111128
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KEYWORD
| nonn,tabl,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Oct 16 2005
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EXTENSIONS
| More terms from Paul Barry (pbarry(AT)wit.ie), Oct 17 2005
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