Search: id:A004070
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%I A004070
%S A004070 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,
%T A004070 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,
%U A004070 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
%N A004070 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.
%C A004070 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
%C A004070 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
%C A004070 Form partial sums across rows of square array of binomial coefficients 
               A026729; see also A008949 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 28 2005
%C A004070 Square array A026729 -> Partial sums across rows
%C A004070 1 0 0 0 0 0 0 . . . . 1 1 1 1 1 1 1 . . . . . .
%C A004070 1 1 0 0 0 0 0 . . . . 1 2 2 2 2 2 2 . . . . . .
%C A004070 1 2 1 0 0 0 0 . . . . 1 3 4 4 4 4 4 . . . . . .
%C A004070 1 3 3 1 0 0 0 . . . . 1 4 7 8 8 8 8 . . . . . .
%C A004070 For other Whitney numbers see A007799.
%D A004070 G. Burosch et al., On posets of m-ary words, Discr. Math., 152 (1996), 
               69-91.
%F A004070 W(n, k) = Sum( binomial(k, i), i=0..n ) - R. W. Gosper
%F A004070 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
%F A004070 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 A004070 Table W(n,k) begins:
%e A004070 1 1 1 1 1 1 1 ...
%e A004070 1 2 3 4 5 6 7 ...
%e A004070 1 2 4 7 11 16 22 ...
%e A004070 1 2 4 8 15 26 42 ...
%o A004070 (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 
               */
%o A004070 /* define the sequence array function for A004070 */ W(n,k) = sum(i=0,
               n,binomial(k,i))
%o A004070 /* visual check ( origin 0,0 ) */ printp(matrix(7,7,n,k,W(n-1,k-1)))
%o A004070 /* 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))","))
%Y A004070 Cf. A007799.
%Y A004070 Rows converge to powers of two (A000079). Subdiagonals include A000225, 
               A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, 
               A035042. Antidiagonal sums are A000071.
%Y A004070 Sequence in context: A140207 A104763 A027751 this_sequence A048887 A047913 
               A117935
%Y A004070 Adjacent sequences: A004067 A004068 A004069 this_sequence A004071 A004072 
               A004073
%K A004070 tabl,nonn,easy,nice
%O A004070 0,5
%A A004070 N. J. A. Sloane (njas(AT)research.att.com).
%E A004070 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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