A002664 -id:A002664 - cp's OEIS frontend

A035040 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,7).

0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 56, 232, 794, 2380, 6476, 16384, 39203, 89846, 199140, 430104, 910596, 1898712, 3913704, 7997952, 16241061, 32828226, 66137152, 132932104, 266752238, 534688516, 1070937812, 2143911424, 4290452423

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
J. Eckhoff, Der Satz von Radon in konvexen Produktstrukturen II, Monat. f. Math., 73 (1969), 7-30.

a(n)= A055248(n, 8). Partial sums of A035039.
Cf. A000079, A000225, A000295, A002663, A002664, A035038-A035042.
Cf. A007318.

a035040 n = a035040_list !! n
a035040_list = map (sum . drop 8) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

a:=n->sum(binomial(n,j),j=8..n): seq(a(n), n=0..32); # Zerinvary Lajos, Jan 04 2007

a=1;lst={};s1=s2=s3=s4=s5=s6=s7=s8=0;Do[s1+=a;s2+=s1;s3+=s2;s4+=s3;s5+=s4;s6+=s5;s7+=s6;s8+=s7;AppendTo[lst,s8];a=a*2,{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Table[2^n-Total[Binomial[n,Range[0,7]]],{n,0,40}] (* Harvey P. Dale, Aug 27 2025 *)

G.f.: x^8/((1-2*x)*(1-x)^8).

a(n) = sum_{k=0..n} C(n, k+8) = sum_{k=8..n} C(n, k); a(n) = 2a(n-1) + C(n-1, 7). - Paul Barry, Aug 23 2004

A035041 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 67, 299, 1093, 3473, 9949, 26333, 65536, 155382, 354522, 784626, 1695222, 3593934, 7507638, 15505590, 31746651, 64574877, 130712029, 263644133, 530396371, 1065084887, 2136022699, 4279934123, 8570386546

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
J. Eckhoff, Der Satz von Radon in konvexen Produktstrukturen II, Monat. f. Math., 73 (1969), 7-30.
Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

a(n)= A055248(n, 9). Partial sums of A035040.
Cf. A000079, A000225, A000295, A002663, A002664, A035038-A035042.
Cf. A007318.

a035041 n = a035041_list !! n
a035041_list = map (sum . drop 9) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

a:=n->sum(binomial(n,j),j=9..n): seq(a(n), n=0..33); # Zerinvary Lajos, Jan 04 2007

a=1;lst={};s1=s2=s3=s4=s5=s6=s7=s8=s9=0;Do[s1+=a;s2+=s1;s3+=s2;s4+=s3;s5+=s4;s6+=s5;s7+=s6;s8+=s7;s9+=s8;AppendTo[lst,s9];a=a*2,{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Table[Sum[ Binomial[n, k], {k, 9, n}], {n, 0, 33}] (* Zerinvary Lajos, Jul 08 2009 *)

G.f.: x^9/((1-2*x)*(1-x)^9).

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 3, 1, 0, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 0, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 0, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 0, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 0

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(0,2n)=T(1,n) by T(0,2n)=T(m,n) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058394, A058395, A057884, (and effectively A007318).

			Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.

Rows are A000035 (A000012 with zeros), A000012, A040000 etc. Columns are A000012, A001477, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863 etc. Diagonals include A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, etc. The triangles A008949, A054143 and A055248 also appear in the half of the array which is not powers of 2.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(1, 1)=1, T(0, 2n)=T(1, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2).

A061290 Square array read by antidiagonals of T(n,k) = T(n-1,k) + T(n-1, floor(k/2)) with T(0,0)=1.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 0, 3, 8, 0, 0, 1, 7, 16, 0, 0, 1, 4, 15, 32, 0, 0, 0, 4, 11, 31, 64, 0, 0, 0, 1, 11, 26, 63, 128, 0, 0, 0, 1, 5, 26, 57, 127, 256, 0, 0, 0, 1, 5, 16, 57, 120, 255, 512, 0, 0, 0, 1, 5, 16, 42, 120, 247, 511, 1024, 0, 0, 0, 0, 5, 16, 42, 99, 247, 502, 1023, 2048, 0, 0

Offset: 0

Henry Bottomley, May 22 2001

Row sums give 3^n.

			T(9,3) = T(8,3) + T(8,floor(3/2)) = T(8,3) + T(8,1) = 247 + 255 = 502. Rows start (1,0,0,0,0,...), (2,1,0,0,0,...), (4,3,1,1,0,...), (8,7,4,4,1,...), etc.

Row sums are A000244. Columns are A000079, A000225, A000295 twice, A002662 four times, A002663 eight times, A002664 sixteen times, A035038 thirty two times, etc.

T(n, k) = C(n, 0) + C(n, 1) + ... + C(n, n-ceiling(log_2(k+1))) = 2^n - C(n, 0) - C(n, 1) - ... - C(n, floor(log_2(k))) = A008949(n, n-A029837(k+1)) = A000079(n) - A008949(n, A000523(k)).

A342380 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^4/24 - x^3/6 - x^2/2 - x - 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 28, 92, 255, 637, 1485, 3301, 7098, 14912, 30826, 63018, 127857, 258095, 519251, 1042379, 2089604, 4185194, 8377704, 16764264, 33539155, 67090961, 134196873, 268411297, 536843070, 1073709892, 2147447190, 4294925846, 8589887653, 17179816227

Offset: 0

Enrique Navarrete, Mar 09 2021

a(n) is the number of binary strings of length n that contain at least five 0's but not all digits are 0.

a(n) is also the number of proper subsets with at least five elements of an n-element set.

			a(9) = 255 since the strings are the 126 permutations of 000001111, the 84 permutations of 000000111, the 36 permutations of 000000011, and the 9 permutations of 000000001.

Index entries for linear recurrences with constant coefficients, signature (7,-20,30,-25,11,-2).

Cf. A342379, A002664.

LinearRecurrence[{7,-20,30,-25,11,-2},{0,0,0,0,0,0,6,28,92,255,637},40] (* Harvey P. Dale, Jun 11 2024 *)

a(n) = 2^n - Sum_{i={0..4,n}} binomial(n,i).

G.f.: x^6*(2*x^4-9*x^3+16*x^2-14*x+6)/((2*x-1)*(x-1)^5). - Alois P. Heinz, Mar 09 2021

A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 4, -1, 1, 11, -5, 1, 1, 26, -16, 6, -1, 1, 57, -42, 22, -7, 1, 1, 120, -99, 64, -29, 8, -1, 1, 247, -219, 163, -93, 37, -9, 1, 1, 502, -466, 382, -256, 130, -46, 10, -1, 1, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2036, -1981, 1816, -1486, 1024

Offset: 0

Philippe Deléham, Nov 30 2013

Row sums are A000079(n) = 2^n.
Diagonal sums are A024493(n+1) = A130781(n).
Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

			Triangle begins:
  1;
  1,    1;
  1,    4,   -1;
  1,   11,   -5,   1;
  1,   26,  -16,   6,   -1;
  1,   57,  -42,  22,   -7,   1;
  1,  120,  -99,  64,  -29,   8,   -1;
  1,  247, -219, 163,  -93,  37,   -9,  1;
  1,  502, -466, 382, -256, 130,  -46, 10,  -1;
  1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;

Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
Cf. A000079, A055248, A024493, A130781, A000302.
Cf. also A003063, A159964, A000012, A001045, A056450, A232015.

G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
|T(2*n,n)| = 4^n = A000302(n).
T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - Werner Schulte, Mar 22 2019

A347017 a(n) = floor(2^(n-1)) - binomial(n,3) + binomial(n,2) - n + 1.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 44, 93, 200, 428, 904, 1883, 3876, 7906, 16020, 32313, 64976, 130392, 261328, 523319, 1047436, 2095822, 4192764, 8386837, 16775192, 33552132, 67106264, 134214803, 268432180, 536867258, 1073737764, 2147479153, 4294962336, 8589929136, 17179863200, 34359731823

Offset: 0

Enrique Navarrete, Aug 10 2021

a(n) is the number of binary strings of length n that contain none, two, five, or a larger odd number of 0's.

			a(6)=22 since the strings are the 15 permutations of 001111, the 6 permutations of 000001, and 111111.

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Cf. A002664, A342379, A342380.

Table[Floor[2^(n-1)]-Binomial[n,3]+Binomial[n,2]-n+1,{n,0,40}] (* or *) LinearRecurrence[{6,-14,16,-9,2},{1,1,2,4,7,12},40] (* Harvey P. Dale, Sep 02 2023 *)

E.g.f.: exp(x)*(sinh(x) + 1 - x + x^2/2 - x^3/6).
From Stefano Spezia, Aug 11 2021: (Start)
O.g.f.: (1 - 5*x + 10*x^2 - 10*x^3 + 4*x^4 + x^5)/((1 - x)^4*(1 - 2*x)).
a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5) for n > 5. (End)

A357255 Triangular array: row n gives the recurrence coefficients for the sequence (c(k) = number of subsets of {1,2,...,n} that have at least k-1 elements) for k >= 1.

Original entry on oeis.org

2, 3, -2, 4, -5, 2, 5, -9, 7, -2, 6, -14, 16, -9, 2, 7, -20, 30, -25, 11, -2, 8, -27, 50, -55, 36, -13, 2, 9, -35, 77, -105, 91, -49, 15, -2, 10, -44, 112, -182, 196, -140, 64, -17, 2, 11, -54, 156, -294, 378, -336, 204, -81, 19, -2

Offset: 1

Clark Kimberling, Sep 24 2022

n-th row sum = 1 for n >= 2.

			First 7 rows:
  2
  3      -2
  4      -5       2
  5      -9       7     -2
  6     -14      16     -9     2
  7     -20      30    -25    11     -2
  8     -27      50    -55    36    -13     2
Row 4 gives recurrence coefficients for the sequence
(r(k)) = (A002662(k)) = (0,0,0,1,5,16,42,99,219,...); i.e.,
r(k) = 5*r(k-1) - 9*r(k-2) + 7*r(k-3) - 2*r(k-4),
with initial values (r(0), r(1), r(2), r(3)) = (0,0,0,1).
(Here r(k) = number of subsets of {1,2,...,4} having at least 3 elements.)

Cf. A029638, A029635.

Cf. sequences generated by recurrences, by row, beginning with row 1: A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039.

Table[Binomial[n, k]*(-1)^(k - 1)*(n + k)/n, {n, 1, 12}, {k, 1, n}]

T(n,k) = (-1)^(k-1) * (C(n,k) + C(n-1,k-1)), for n >= 1, k >= 1.

T(n,k) = (-1)^(k-1) * C(n,k)*(n+k)/n, for n >= 1, k >= 1.

cp's OEIS Frontend

A035040 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,7).

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A035041 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,8).

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A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

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A061290 Square array read by antidiagonals of T(n,k) = T(n-1,k) + T(n-1, floor(k/2)) with T(0,0)=1.

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A342380 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^4/24 - x^3/6 - x^2/2 - x - 1).

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A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

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A347017 a(n) = floor(2^(n-1)) - binomial(n,3) + binomial(n,2) - n + 1.

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A357255 Triangular array: row n gives the recurrence coefficients for the sequence (c(k) = number of subsets of {1,2,...,n} that have at least k-1 elements) for k >= 1.

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