|
%I #176 Dec 11 2022 11:49:33
%S 1,1,2,1,3,4,1,4,7,8,1,5,11,15,16,1,6,16,26,31,32,1,7,22,42,57,63,64,
%T 1,8,29,64,99,120,127,128,1,9,37,93,163,219,247,255,256,1,10,46,130,
%U 256,382,466,502,511,512,1,11,56,176,386,638,848,968,1013,1023,1024,1,12,67,232,562,1024,1486,1816,1981,2036,2047,2048
%N Triangle read by rows of partial sums of binomial coefficients: T(n,k) = Sum_{i=0..k} binomial(n,i) (0 <= k <= n); also dimensions of Reed-Muller codes.
%C The second-left-from-middle column is A000346: T(2n+2, n) = A000346(n). - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
%C T(n,k) is the maximal number of regions into which n hyperplanes of co-dimension 1 divide R^k (the Cake-Without-Icing numbers). - _Rob Johnson_, Jul 27 2008
%C T(n,k) gives the number of vertices within distance k (measured along the edges) of an n-dimensional unit cube, (i.e., the number of vertices on the hypercube graph Q_n whose distance from a reference vertex is <= k). - _Robert Munafo_, Oct 26 2010
%C A triangle formed like Pascal's triangle, but with 2^n for n >= 0 on the right border instead of 1. - _Boris Putievskiy_, Aug 18 2013
%C For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 04 2013
%C Consider each "1" as an apex of two sequences: the first is the set of terms in the same row as the "1", but the rightmost term in the row repeats infinitely. Example: the row (1, 4, 7, 8) becomes (1, 4, 7, 8, 8, 8, ...). The second sequence begins with the same "1" but is the diagonal going down and to the right, thus: (1, 5, 16, 42, 99, 219, 466, ...). It appears that for all such sequence pairs, the binomial transform of the first, (1, 4, 7, 8, 8, 8, ...) in this case; is equal to the second: (1, 5, 16, 42, 99, ...). - _Gary W. Adamson_, Aug 19 2015
%C Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let q(n) be the sum of polynomials in the n-th generation of T*. For n >= 0, row n of A008949 gives the coefficients of q(n+1); e.g., (row 3) = (1, 4, 7, 8) matches x^3 + 4*x^2 + 7*x + 9, which is the sum of the 8 polynomials in the 4th generation of T*. - _Clark Kimberling_, Jun 16 2016
%C T(n,k) is the number of subsets of [n]={1,...,n} of at most size k. Equivalently, T(n,k) is the number of subsets of [n] of at least size n-k. Counting the subsets of at least size (n-k) by conditioning on the largest element m of the smallest (n-k) elements of such a subset provides the formula T(n,k) = Sum_{m=n-k..n} C(m-1,n-k-1)*2^(n-m), and, by letting j=m-n+k, we obtain T(n,k) = Sum_{j=0..k} C(n+j-k-1,j)*2^(k-j). - _Dennis P. Walsh_, Sep 25 2017
%C If the interval of integers 1..n is shifted up or down by k, making the new interval 1+k..n+k or 1-k..n-k, then T(n-1,n-1-k) (= 2^(n-1)-T(n-1,k-1)) is the number of subsets of the new interval that contain their own cardinal number as an element. - _David Pasino_, Nov 01 2018
%C This triangle is also called Bernoulli's triangle. - _Robert FERREOL_, Oct 11 2022
%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 376.
%H Harvey P. Dale, <a href="/A008949/b008949.txt">Table of n, a(n) for n = 0..10000</a>
%H Milica Andelic, C. M. da Fonseca and A. Pereira, <a href="https://arxiv.org/abs/1609.04208">The mu-permanent, a new graph labeling, and a known integer sequence</a>, arXiv:1609.04208 [math.CO], 2016.
%H Rob Johnson, <a href="http://web.archive.org/web/20140822210215/http://www.whim.org/nebula/math/spacediv.html">Dividing Space</a>.
%H Norman Lindquist and Gerard Sierksma, <a href="https://doi.org/10.1016/0097-3165(81)90015-7">Extensions of set partitions</a>, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I.
%H Denis Neiter and Amsha Proag, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Proag/proag3.html">Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
%H Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/subcount.pdf">A note on counting subsets of restricted size</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F From partial sums across rows of Pascal triangle A007318.
%F T(n, 0) = 1, T(n, n) = 2^n, T(n, k) = T(n-1, k-1) + T(n-1, k), 0 < k < n.
%F G.f.: (1 - x*y)/((1 - y - x*y)*(1 - 2*x*y)). - Antonio Gonzalez (gonfer00(AT)gmail.com), Sep 08 2009
%F T(2n,n) = A032443(n). - _Philippe Deléham_, Sep 16 2009
%F T(n,k) = 2 T(n-1,k-1) + binomial(n-1,k) = 2 T(n-1,k) - binomial(n-1,k). - _M. F. Hasler_, May 30 2010
%F T(n,k) = binomial(n,n-k)* 2F1(1, -k; n+1-k; -1). - _Olivier Gérard_, Aug 02 2012
%F For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 18 2013
%F T(n,floor(n/2)) = A027306(n). - _Reinhard Zumkeller_, Nov 14 2014
%F T(n,n) = 2^n, otherwise for 0 <= k <= n-1, T(n,k) = 2^n - T(n,n-k-1). - _Bob Selcoe_, Mar 30 2017
%F For fixed j >= 0, lim_{n -> oo} T(n+1,n-j+1)/T(n,n-j) = 2. - _Bob Selcoe_, Apr 03 2017
%F T(n,k) = Sum_{j=0..k} C(n+j-k-1,j)*2^(k-j). - _Dennis P. Walsh_, Sep 25 2017
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 3, 4;
%e 1, 4, 7, 8;
%e 1, 5, 11, 15, 16;
%e 1, 6, 16, 26, 31, 32;
%e 1, 7, 22, 42, 57, 63, 64;
%e 1, 8, 29, 64, 99, 120, 127, 128;
%e 1, 9, 37, 93, 163, 219, 247, 255, 256;
%e 1, 10, 46, 130, 256, 382, 466, 502, 511, 512;
%e 1, 11, 56, 176, 386, 638, 848, 968, 1013, 1023, 1024;
%e ...
%p A008949 := proc(n,k) local i; add(binomial(n,i),i=0..k) end; # Typo corrected by _R. J. Mathar_, Oct 26 2010
%t Table[Length[Select[Subsets[n], (Length[ # ] <= k) &]], {n, 0, 12}, {k, 0, n}] // Grid (* _Geoffrey Critzer_, May 13 2009 *)
%t Flatten[Accumulate/@Table[Binomial[n,i],{n,0,20},{i,0,n}]] (* _Harvey P. Dale_, Aug 08 2015 *)
%t T[ n_, k_] := If[ n < 0 || k > n, 0, Binomial[n, k] Hypergeometric2F1[1, -k, n + 1 - k, -1]; (* _Michael Somos_, Aug 05 2017 *)
%o (PARI) A008949(n)=T8949(t=sqrtint(2*n-sqrtint(2*n)),n-t*(t+1)/2)
%o T8949(r,c)={ 2*c > r || return(sum(i=0,c,binomial(r,i))); 1<<r - sum( i=c+1,r,binomial(r,i))} \\ _M. F. Hasler_, May 30 2010
%o (PARI) {T(n, k) = if(k>n, 0, sum(i=0, k, binomial(n, i)))}; /* _Michael Somos_, Aug 05 2017 */
%o (Haskell)
%o a008949 n k = a008949_tabl !! n !! k
%o a008949_row n = a008949_tabl !! n
%o a008949_tabl = map (scanl1 (+)) a007318_tabl
%o -- _Reinhard Zumkeller_, Nov 23 2012
%o (GAP) T:=Flat(List([0..11],n->List([0..n],k->Sum([0..k],j->Binomial(n+j-k-1,j)*2^(k-j))))); # _Muniru A Asiru_, Nov 25 2018
%o (Magma) [[(&+[Binomial(n,j): j in [0..k]]): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Nov 25 2018
%o (Sage) [[sum(binomial(n,j) for j in range(k+1)) for k in range(n+1)] for n in range(12)] # _G. C. Greubel_, Nov 25 2018
%Y Diagonals are given by A000079, A000225, A000295, A002662, A002663, A002664, A035038-A035042.
%Y Columns are given by A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863. - _Ken Shirriff_, Jun 28 2011
%Y Row sums sequence is A001792.
%Y T(n, m)= A055248(n, n-m).
%Y Cf. A110555, A007318, A000346, A171886, A228196, A228576.
%Y Cf. also A027306, A249111, A163866, A261363.
%Y Cf. A000071, A001924
%K tabl,nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Mar 23 2000
|
