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Search: id:A048004
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| A048004 |
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Triangular array read by rows: T(n,k)=number of binary words of length n whose longest run of consecutive 1's has length k, for n >= 0, 0 <= k <= n. |
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+0
8
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| 1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 7, 5, 2, 1, 1, 12, 11, 5, 2, 1, 1, 20, 23, 12, 5, 2, 1, 1, 33, 47, 27, 12, 5, 2, 1, 1, 54, 94, 59, 28, 12, 5, 2, 1, 1, 88, 185, 127, 63, 28, 12, 5, 2, 1, 1, 143, 360, 269, 139, 64, 28, 12, 5, 2, 1, 1, 232, 694, 563, 303, 143, 64, 28, 12, 5, 2, 1, 1, 376, 1328
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Number of compositions of n+1 having largest part equal to k+1. Example: T(4,2)=5 because we have 3+2, 2+3, 3+1+1, 1+3+1, and 1+1+3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005
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REFERENCES
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J. Kappraff, Beyond Measure, World Scientific, 2002; see pp. 471-472.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 155.
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FORMULA
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T(n, k) = 0 if k < 0 or k > n, 1 if k = 0 or k = n, 2T(n-1, k)+T(n-1, k-1)-2T(n-2, k-1)+T(n-k-1, k-1)-T(n-k-2, k) otherwise - David W. Wilson
T(n, k) = A048887(n+1, k+1)-A048887(n+1, k) - Henry Bottomley (se16(AT)btinternet.com), Oct 29 2002
G.f. for column k=(1-x)^2*x^k/[(1-2x+x^(k+1))*(1-2x+x^(k+2))]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005
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EXAMPLE
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Rows: {1}; {1,1}; {1,2,1}; {1,4,2,1}; ...
Example: T(4,2)=5 because we have 0011,0110,1011,1100, and 1101.
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MAPLE
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G:=k->(1-x)^2*x^k/(1-2*x+x^(k+1))/(1-2*x+x^(k+2)): for k from 0 to 14 do g[k]:=series(G(k), x=0, 15) od: 1, seq(seq(coeff(g[k], x^n), k=0..n), n=1..12); (Deutsch)
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CROSSREFS
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See A126198 and A048887 for closely related arrays.
T(n, 2)=Fibonacci(n+2)-1, A000071, T(n, 3)=b(n) for n=3, 4, ..., where b=A000100, T(n, 4)=c(n) for n=4, 5, ..., where c=A000102.
Nonnegative elements of columns approach A045623. Cf. A048003.
Sequence in context: A098063 A106396 A140998 this_sequence A114394 A059623 A140997
Adjacent sequences: A048001 A048002 A048003 this_sequence A048005 A048006 A048007
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005
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