OFFSET
0,3
COMMENTS
The second-left-from-middle column is A000346: T(2n+2, n) = A000346(n). - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
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
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
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
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013
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
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
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
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
This triangle is also called Bernoulli's triangle. - Robert FERREOL, Oct 11 2022
REFERENCES
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 376.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..10000
Milica Andelic, C. M. da Fonseca and A. Pereira, The mu-permanent, a new graph labeling, and a known integer sequence, arXiv:1609.04208 [math.CO], 2016.
Stefan Forcey, Planes and axioms, Univ. Akron (2024). See p. 2.
Rob Johnson, Dividing Space.
Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Dennis P. Walsh, A note on counting subsets of restricted size
FORMULA
From partial sums across rows of Pascal triangle A007318.
T(n, 0) = 1, T(n, n) = 2^n, T(n, k) = T(n-1, k-1) + T(n-1, k), 0 < k < n.
G.f.: (1 - x*y)/((1 - y - x*y)*(1 - 2*x*y)). - Antonio Gonzalez (gonfer00(AT)gmail.com), Sep 08 2009
T(2n,n) = A032443(n). - Philippe Deléham, Sep 16 2009
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
T(n,k) = binomial(n,n-k)* 2F1(1, -k; n+1-k; -1). - Olivier Gérard, Aug 02 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
T(n,floor(n/2)) = A027306(n). - Reinhard Zumkeller, Nov 14 2014
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
For fixed j >= 0, lim_{n -> oo} T(n+1,n-j+1)/T(n,n-j) = 2. - Bob Selcoe, Apr 03 2017
T(n,k) = Sum_{j=0..k} C(n+j-k-1,j)*2^(k-j). - Dennis P. Walsh, Sep 25 2017
EXAMPLE
Triangle begins:
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;
1, 8, 29, 64, 99, 120, 127, 128;
1, 9, 37, 93, 163, 219, 247, 255, 256;
1, 10, 46, 130, 256, 382, 466, 502, 511, 512;
1, 11, 56, 176, 386, 638, 848, 968, 1013, 1023, 1024;
...
MAPLE
A008949 := proc(n, k) local i; add(binomial(n, i), i=0..k) end; # Typo corrected by R. J. Mathar, Oct 26 2010
MATHEMATICA
Table[Length[Select[Subsets[n], (Length[ # ] <= k) &]], {n, 0, 12}, {k, 0, n}] // Grid (* Geoffrey Critzer, May 13 2009 *)
Flatten[Accumulate/@Table[Binomial[n, i], {n, 0, 20}, {i, 0, n}]] (* Harvey P. Dale, Aug 08 2015 *)
T[ n_, k_] := If[ n < 0 || k > n, 0, Binomial[n, k] Hypergeometric2F1[1, -k, n + 1 - k, -1]; (* Michael Somos, Aug 05 2017 *)
PROG
(PARI) A008949(n)=T8949(t=sqrtint(2*n-sqrtint(2*n)), n-t*(t+1)/2)
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
(PARI) {T(n, k) = if(k>n, 0, sum(i=0, k, binomial(n, i)))}; /* Michael Somos, Aug 05 2017 */
(Haskell)
a008949 n k = a008949_tabl !! n !! k
a008949_row n = a008949_tabl !! n
a008949_tabl = map (scanl1 (+)) a007318_tabl
-- Reinhard Zumkeller, Nov 23 2012
(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
(Magma) [[(&+[Binomial(n, j): j in [0..k]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 25 2018
(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
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 23 2000
STATUS
approved