A249111 -id:A249111 - cp's OEIS frontend

A008949 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.

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, 1, 12, 67, 232, 562, 1024, 1486, 1816, 1981, 2036, 2047, 2048

Offset: 0

N. J. A. Sloane

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

			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;
  ...

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 376.

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.
Stefan Forcey, Counting plane arrangements via oriented matroids, arXiv:2504.11461 [math.HO], 2025. See p. 18.
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
Wikipedia, Bernoulli's triangle
Index entries for triangles and arrays related to Pascal's triangle

Diagonals are given by A000079, A000225, A000295, A002662, A002663, A002664, A035038-A035042.
Columns are given by A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863. - Ken Shirriff, Jun 28 2011
Row sums sequence is A001792.
T(n, m)= A055248(n, n-m).
Cf. A110555, A007318, A000346, A171886, A228196, A228576.
Cf. also A027306, A249111, A163866, A261363.
Cf. A000071, A001924

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

a008949 n k = a008949_tabl !! n !! k
a008949_row n = a008949_tabl !! n
a008949_tabl = map (scanl1 (+)) a007318_tabl
-- Reinhard Zumkeller, Nov 23 2012

[[(&+[Binomial(n,j): j in [0..k]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 25 2018

A008949 := proc(n,k) local i; add(binomial(n,i),i=0..k) end; # Typo corrected by R. J. Mathar, Oct 26 2010

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 *)

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<M. F. Hasler, May 30 2010

{T(n, k) = if(k>n, 0, sum(i=0, k, binomial(n, i)))}; /* Michael Somos, Aug 05 2017 */

row(n) = my(v=vector(n+1, k, binomial(n,k-1))); vector(#v, k, sum(i=1, k, v[i])); \\ Michel Marcus, Apr 13 2025

[[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

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

More terms from Larry Reeves (larryr(AT)acm.org), Mar 23 2000

A249095 Triangle read by rows: interleaving successive pairs of rows of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 1, 5, 4, 10, 6, 10, 4, 5, 1, 1, 1, 1, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1, 1, 1, 1, 7, 6, 21, 15, 35, 20, 35, 15, 21, 6, 7, 1, 1, 1, 1, 8, 7, 28, 21, 56, 35, 70, 35, 56, 21, 28, 7, 8, 1, 1

Offset: 0

Reinhard Zumkeller, Nov 14 2014

Length of row n = 2*n+1;
T(n,2*k) = A007318(n,k), 0 <= k <= n;
T(n,2*k+1) = A007318(n-1,k-1), n > 0 and 0 <= k < n;
T(n,k) = T(n-1,k-2) + T(n-1,k), n > 0 and 2 <= k <= n-2;
T(n,2*k) = T(n-1,2*k) + T(n-1,2*(k-1)), k = 0..n;
T(n,2*k+1) = T(n-2,2*k), k = 0..n-1;
T(n,n) = A128014(n);
A105321(n) = number of odd terms in row n;
A249304(n) = number of even terms in row n;
T(n,k) mod 2 = A249133(n,k).

			The triangle begins:
.  0:                              1
.  1:                           1  1   1
.  2:                       1   1  2   1  1
.  3:                    1  1   3  2   3  1  1
.  4:                 1  1  4   3  6   3  4  1  1
.  5:              1  1  5  4  10  6  10  4  5  1  1
.  6:           1  1  6  5 15  10 20  10 15  5  6  1  1
.  7:        1  1  7  6 21 15  35 20  35 15 21  6  7  1  1
.  8:     1  1  8  7 28 21 56  35 70  35 56 21 28  7  8  1  1
.  9:  1  1  9  8 36 28 84 56 126 70 126 56 84 28 36  8  9  1  1 .

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A005408 (row lengths), A128014 (central terms), A003945 (row sums), A249111 (partial sums per row), A007318 (Pascal).

Cf. A105321, A249304, A249307, A249133.

import Data.List (transpose)
a249095 n k = a249095_tabf !! n !! k
a249095_row n = a249095_tabf !! n
a249095_tabf = [1] : map (concat . transpose)
   (zipWith ((. return) . (:)) (tail a007318_tabl) a007318_tabl)

t[n_, k_] := If[n > 1 && 1 < k < 2*n - 1, If[EvenQ[k], t[n - 1, k] + t[n - 1, k - 2], t[n - 1, k - 1]], 1]; Grid[Table[t[n, k], {n, 0, 9}, {k, 0, 2*n}]] (* L. Edson Jeffery, Nov 30 2014 *)

T(n,2*k) = T(n,2*k-1) + T(n,2*k+1), 0 < k < n.

A128543 a(n) = floor(2^(n-2)3n).

Original entry on oeis.org

1, 6, 18, 48, 120, 288, 672, 1536, 3456, 7680, 16896, 36864, 79872, 172032, 368640, 786432, 1671168, 3538944, 7471104, 15728640, 33030144, 69206016, 144703488, 301989888, 629145600, 1308622848, 2717908992, 5637144576

Offset: 1

Gary W. Adamson, Mar 10 2007

Also row sums of triangle A249111. - Reinhard Zumkeller, Nov 15 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A007310, A134239, A249111.

Concatenation([1], List([2..40], n-> 3*n*2^(n-2))); # G. C. Greubel, Jul 11 2019

a128543 = sum . a134239_row . subtract 1
-- Reinhard Zumkeller, Nov 15 2014

I:=[1, 6, 18]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012

CoefficientList[Series[(1+2*x-2*x^2)/(1-2*x)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 28 2012 *)

a(n)=3*n*2^n\4 \\ Charles R Greathouse IV, Oct 07 2015

[1]+[3*n*2^(n-2) for n in (2..40)] # G. C. Greubel, Jul 11 2019

Binomial transform of A007310 (assuming offset 0 in both sequences).
Row sums of triangle A134239. - Gary W. Adamson, Oct 14 2007
a(n) = 3*n*2^(n-2) for n>1. - R. J. Mathar, Oct 25 2011
From Colin Barker, Mar 22 2012: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) for n>3.
G.f.: x*(1+2*x-2*x^2)/(1-2*x)^2. (End)

Definition corrected by M. F. Hasler, Nov 05 2014

cp's OEIS Frontend

A008949 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.

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A249095 Triangle read by rows: interleaving successive pairs of rows of Pascal's triangle.

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A128543 a(n) = floor(2^(n-2)*3*n).

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A248574 a(n) = A027306(n) + A027306(n-1) for n > 0; a(0) = 1.

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A128543 a(n) = floor(2^(n-2)3n).