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A119900
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Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs (0<=k<=n; for example, the binary word 1/0/01/01/1/1/01 has 7 strictly increasing runs).
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9
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1, 0, 2, 0, 1, 3, 0, 0, 4, 4, 0, 0, 1, 10, 5, 0, 0, 0, 6, 20, 6, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 1, 55, 330, 462, 165, 11, 0, 0, 0, 0, 0, 0, 12, 220, 792, 792, 220, 12, 0, 0, 0, 0, 0, 0, 1, 78
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Sum of entries in row n is 2^n (A000079). Sum of entries in column k is A001906(k+1) (the even indexed Fibonacci numbers). Row n contains 1+floor(n/2) nonzero terms. Sum(k*T(n,k),k=0..n)=(3n+1)*2^(n-2)=A066373(n+1) for n>=1. T(n,k)=A034867(n,n-k).
Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,...] DELTA [2,-1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
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FORMULA
| T(n,k)=binom(n+1,2k-n). G.f.=1/[1-2tz-t(1-t)z^2].
Contribution from Tom Copeland, Sep 30 2011: (Start)
With K(x,t) = 1/{d/dx{x/[t-1+1/(1-x)]}} = [t-1+1/(1-x)]^2/{t-[x/(1-x)]^2}, the G.f. of A119900 = K(x*t,t)-t+1.
From formulas in A134264: K(x,t)d/dx is a generator for A001263. A refinement of A119900 to partition polynomials is given by umbralizing
K(x,t) roughly as K(h.x,h_0) and precisely as in A134264 as
W(x)= 1/{d/dx[f(x)]}=1/{d/dx[x/h(x)]}. (End)
T(n,k)=2*T(n-1,k-1)+T(n-2,k-1)-T(n-2,k-2). - DELEHAM Philippe, 0ct 02 2011
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EXAMPLE
| T(5,3)=6 because we have 0/01/01, 01/0/01, 01/01/0, 01/1/01, 01/01/1 and 1/01/01 (the runs are separated by /).
Triangle starts:
1;
0,2;
0,1,3;
0,0,4,4;
0,0,1,10,5;
0,0,0,6,20,6;
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MAPLE
| T:=(n, k)->binomial(n+1, 2*k-n): for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A000079, A001906, A066373, A034867.
Cf. A098158 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
Sequence in context: A132213 A202502 A154312 * A141097 A096335 A191910
Adjacent sequences: A119897 A119898 A119899 * A119901 A119902 A119903
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), May 27 2006
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EXTENSIONS
| keyword: tabl added by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 26 2010
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