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A004070
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Table of Whitney numbers W(n,k) read by antidiagonals, where W(n,k) is maximal number of pieces into which n-space is sliced by k hyperplanes, n >= 0, k >= 0.
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24
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 2, 4, 7, 5, 1, 1, 2, 4, 8, 11, 6, 1, 1, 2, 4, 8, 15, 16, 7, 1, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 1, 2, 4, 8, 16, 32, 64, 120, 163
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| As a number triangle, this is given by T(n,k)=sum{j=0..n, C(n,j)(-1)^(n-j)sum{i=0..j, C(j+k,i-k)}}. - Paul Barry (pbarry(AT)wit.ie), Aug 23 2004
As a number triangle, this is the Riordan array (1/(1-x), x(1+x)) with T(n,k)=sum{i=0..n, binomial(k,i-k)}. Diagonal sums are then A023434(n+1). - Paul Barry (pbarry(AT)wit.ie), Feb 16 2005
Form partial sums across rows of square array of binomial coefficients A026729; see also A008949 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 28 2005
Square array A026729 -> Partial sums across rows
1 0 0 0 0 0 0 . . . . 1 1 1 1 1 1 1 . . . . . .
1 1 0 0 0 0 0 . . . . 1 2 2 2 2 2 2 . . . . . .
1 2 1 0 0 0 0 . . . . 1 3 4 4 4 4 4 . . . . . .
1 3 3 1 0 0 0 . . . . 1 4 7 8 8 8 8 . . . . . .
For other Whitney numbers see A007799.
W(n,k) is the number of length k binary sequences containing no more than n 1's. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 15 2010]
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 15 2010: (Start)
Viewed as a number triangle, T(n,k) is the number of internal nodes of the Fibonacci tree of order n+2 at level k. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
(End)
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REFERENCES
| G. Burosch et al., On posets of m-ary words, Discr. Math., 152 (1996), 69-91.
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 15 2010]
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 15 2010]
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FORMULA
| W(n, k) = Sum( binomial(k, i), i=0..n ) - R. W. Gosper
W(n, k)=if k=0 or n=0 then 1 else W(n, k-1)+W(n-1, k-1) - David Broadhurst (D.Broadhurst(AT)open.ac.uk), Jan 05 2000
The table W(n,k) = A000012 * A007318(transform), where A000012 = (1; 1,1; 1,1,1;...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2007
E.g.f. for row n is: (1 + x + x^2/2! + ... + x^n/n!)* exp(x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 15 2010]
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EXAMPLE
| Table W(n,k) begins:
1 1 1 1 1 1 1 ...
1 2 3 4 5 6 7 ...
1 2 4 7 11 16 22 ...
1 2 4 8 15 26 42 ...
W(2,4) = 11 because there are 11 length 4 binary sequences containing no more than 2 1's: {0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}, {0, 0, 1, 1}, {0, 1, 0, 0}, {0, 1, 0, 1}, {0, 1, 1, 0}, {1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 1, 0, 0}. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 15 2010]
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MATHEMATICA
| Transpose[ Table[Table[Sum[Binomial[n, k], {k, 0, m}], {m, 0, 15}], {n, 0, 15}]] // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 15 2010]
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PROG
| (PARI) /* array read by antidiagonals up coordinate ( origin 0 -> 0, 0 ) index functions */ t1(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1) /* A025581 */ t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2) /* A002262 */
/* define the sequence array function for A004070 */ W(n, k) = sum(i=0, n, binomial(k, i))
/* visual check ( origin 0, 0 ) */ printp(matrix(7, 7, n, k, W(n-1, k-1)))
/* print the sequence entries by antidiagonals going up ( origin 0, 0 ) */ print1("S A004070 "); for(n=0, 32, print1(W(t1(n), t2(n))", ")); print1("T A004070 "); for(n=33, 61, print1(W(t1(n), t2(n))", ")); print1("U A004070 "); for(n=62, 86, print1(W(t1(n), t2(n))", "))
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CROSSREFS
| Cf. A007799.
Rows converge to powers of two (A000079). Subdiagonals include A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, A035042. Antidiagonal sums are A000071.
Rows are: A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 15 2010]
A178522, A178524 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 15 2010]
Sequence in context: A104763 A027751 A181322 * A180562 A199711 A048887
Adjacent sequences: A004067 A004068 A004069 * A004071 A004072 A004073
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KEYWORD
| tabl,nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000. Additional comments and PARI code from Michael Somos, Apr 28, 2000.
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