A002663 -id:A002663 - cp's OEIS frontend

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

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).

A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 11, 15, 15, 11, 5, 1, 16, 26, 30, 26, 16, 6, 1, 22, 42, 56, 56, 42, 22, 7, 1, 29, 64, 98, 112, 98, 64, 29, 8, 1, 37, 93, 162, 210, 210, 162, 93, 37, 9, 1, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10, 1, 56, 176, 385, 627, 792, 792, 627

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 05 2006

Binomial transform of the infinite tridiagonal matrix with main diagonal, (1,1,1,...), subdiagonal, (0,0,0,...) and subsubdiagonal, (1,1,1,...). Sum of entries in row n = 2^(n+1) - n - 1 = A000325(n+1).

Riordan array ((1-2x+2x^2)/(1-x)^3, x/(1-x)). - Paul Barry, Apr 08 2011

			Row 3 = (4, 4, 3, 1), then row 4 = (7, 8, 7, 4, 1).
First few rows of the triangle are
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   7,  8,  7,  4,  1;
  11, 15, 15, 11,  5,  1;
  16, 26, 30, 26, 16,  6,  1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins
   1, 1;
   1, 1, 1;
   0, 0, 1, 1;
  -1, 0, 0, 1, 1;
   0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 1, 1;
  -1, 0, 0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
(End)

Cf. A098574, A000125, A055795, A027660, A055796, A002663.

Cf. A000325.

T:=(n,k)->binomial(n,k)+binomial(n,k+2): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Flatten[Table[Binomial[n,k]+Binomial[n,k+2],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 12 2015 *)

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 12 2014

Edited by N. J. A. Sloane, Nov 29 2006

A342379 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^3/6 - x^2/2 - x - 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 21, 63, 162, 381, 847, 1815, 3796, 7813, 15913, 32191, 64838, 130237, 261155, 523127, 1047224, 2095589, 4192509, 8386559, 16774890, 33551805, 67105911, 134214423, 268431772, 536866821, 1073737297, 2147478655, 4294961806, 8589928573, 17179862603

Offset: 0

Enrique Navarrete, Mar 09 2021

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

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

			a(7) = 63 since the strings are the 35 permutations of 0000111, the 21 permutations of 0000011, and the 7 permutations of 0000001.

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

Cf. A342352, A002663.

a(n) = 2^n - Sum_{i={0..3,n}} binomial(n,i).

G.f.: x^5*(2*x^3-7*x^2+9*x-5)/((2*x-1)*(x-1)^4). - Alois P. Heinz, Mar 09 2021

A347435 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6) ).

Original entry on oeis.org

1, 0, 0, 0, 1, 6, 22, 64, 198, 1138, 10004, 83920, 617993, 4226028, 30103686, 251883012, 2490287821, 26456763078, 281404300348, 2966101610920, 31877462564554, 362624252399566, 4437794875670072, 57612897938229380, 773900876490016325, 10599854900351622752

Offset: 0

Ilya Gutkovskiy, Sep 02 2021

nonn

Exponential transform of A002663.

Cf. A002663, A055882, A057837, A143405, A347432, A347434.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j^3/6-5*j/6-1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x - x^2/2 - x^3/6)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - 1 - k (k^2 + 5)/6) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002663(k) * a(n-k).

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)).

A106471 A number triangle with duplicated columns of the form 2^n - Sum_{j=0..2k-1} C(n,j).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 4, 1, 16, 8, 11, 4, 1, 32, 16, 26, 11, 6, 1, 64, 32, 57, 26, 22, 6, 1, 128, 64, 120, 57, 64, 22, 8, 1, 256, 128, 247, 120, 163, 64, 37, 8, 1, 512, 256, 502, 247, 382, 163, 130, 37, 10, 1, 1024, 512, 1013, 502, 848, 382, 386, 130, 56, 10, 1, 2048, 1024

Offset: 0

Paul Barry, May 03 2005

Columns include A000079, A000295, A002663, A035038, A035040.
Row sums are A106472.
Product of binomial matrix binomial(n,k) and number triangle A106465.

			Triangle begins
   1;
   2,  1;
   4,  2,  1;
   8,  4,  4,  1;
  16,  8, 11,  4,  1;
  32, 16, 26, 11,  6, 1;
  64, 32, 57, 26, 22, 6, 1;

Column 2k has g.f. x^(2*k)/((1-2*x)*(1-x)^(2*k-2)).

Column 2k+1 has g.f. x^(2*k+1)/((1-2*x)*(1-x)^(2*k)).

A124326 T(n,m) = A007318(n,m) - A077028(n,m).

Original entry on oeis.org

1, 3, 3, 6, 10, 6, 10, 22, 22, 10, 15, 40, 53, 40, 15, 21, 65, 105, 105, 65, 21, 28, 98, 185, 226, 185, 98, 28, 36, 140, 301, 431, 431, 301, 140, 36, 45, 192, 462, 756, 887, 756, 462, 192, 45, 55, 255, 678, 1246, 1673, 1673, 1246, 678, 255, 55, 66, 330, 960, 1956, 2954

Offset: 1

Roger L. Bagula, Jun 26 2007

First term of n-th row is n*(n+1)/2.
Row sum are A002663 (without initial zeros).
Appears to be the triangle resulting from adding the row number (first row numbered 0) of Pascal's triangle (A007318) to each entry in that row, subtracting the corresponding entries in the triangle formed by taking the finite diagonals in the multiplication table in order of increasing length (A003991), and removing the outer two layers, which consist entirely of 0's.
Each value of the sequence T(x,y) is equal to the sum of all values in A014430 that are in the rectangle defined by the tip (0,0) and the position (x,y). - Jon Perry, Sep 11 2013

			Table begins
   1;
   3,   3;
   6,  10,   6;
  10,  22,  22,  10;
  15,  40,  53,  40,  15;
  21,  65, 105, 105,  65,  21;
  28,  98, 185, 226, 185,  98,  28;
  36, 140, 301, 431, 431, 301, 140,  36;
  45, 192, 462, 756, 887, 756, 462, 192,  45;
  ...
If the zeros are included, the table begins
   0;
   0, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 1, 0, 0;
   0, 0, 3, 3, 0, 0;
   0, 0, 6, 10, 6, 0, 0;
   0, 0, 10, 22, 22, 10, 0, 0;
   ... - from _Michael De Vlieger_, May 27 2024

Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10, also Glasnik Matematički, (2024) Vol. 59, No. 79, 313-325. See pp. 319-320.

Cf. A077028, A007318, A002663, A014430.

a = Table[Flatten[Table[If[Binomial[m, n] - (1 +n (m - n)) == 0, {}, Binomial[m, n] - (1 + n (m - n))], {n, 0, m}]], {m, 0, 14}]

T(n,m) = A007318(n,m) - A077028(n,m) (skipping zeros).

A135552 Riordan array (1/((1-2x)(1-x)^2), -x/(1-x)^2).

Original entry on oeis.org

1, 4, -1, 11, -6, 1, 26, -22, 8, -1, 57, -64, 37, -10, 1, 120, -163, 130, -56, 12, -1, 247, -382, 386, -232, 79, -14, 1, 502, -848, 1024, -794, 378, -106, 16, -1, 1013, -1816, 2510, -2380, 1471, -576, 137, -18, 1, 2036, -3797, 5812, -6476, 4944, -2517, 834, -172, 20, -1, 4083, -7814, 12911, -16384, 14893, -9402, 4048, -1160, 211, -22, 1

Offset: 1

Roger L. Bagula, Apr 08 2008

Row sums are A024495. First column is essentially A000295, second column is essentially -A002663. - Ralf Stephan, Jan 03 2014

			{1},
{4, -1},
{11, -6, 1},
{26, -22, 8, -1},
{57, -64, 37, -10, 1},
{120, -163, 130, -56, 12, -1},
{247, -382, 386, -232, 79, -14, 1},
{502, -848, 1024, -794, 378, -106, 16, -1},
{1013, -1816, 2510, -2380, 1471, -576, 137, -18, 1},
{2036, -3797, 5812, -6476, 4944, -2517, 834, -172, 20, -1},
{4083, -7814, 12911, -16384, 14893, -9402, 4048, -1160, 211, -22, 1}

G. C. Greubel, Table of n, a(n) for the first 25 rows

Cf. A136674, A109954.

Clear[p, x, n, a]; p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = 4 - x; p[x_, n_] := p[x, n] = (4 - x)*p[x, n - 1] + (-5 + 2*x)*p[x, n - 2] + 2*p[x, n - 3]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

Coefficients of polynomials defined by recurrence: c0=2; p(x, n) = (2 + c0 - x)*p(x, n - 1) + (-1 - c0 (2 - x))*p(x, n - 2) + c0*p(x, n - 3).

New name from Ralf Stephan, Jan 03 2014

A216414 a(n) = (-1)^(n-3)*binomial(n,3) - 1.

Original entry on oeis.org

0, -5, 9, -21, 34, -57, 83, -121, 164, -221, 285, -365, 454, -561, 679, -817, 968, -1141, 1329, -1541, 1770, -2025, 2299, -2601, 2924, -3277, 3653, -4061, 4494, -4961, 5455, -5985, 6544, -7141, 7769, -8437, 9138, -9881, 10659, -11481, 12340, -13245, 14189

Offset: 3

V. Raman, Sep 07 2012

a(n-1) gives the number of times the sum of all numbers divisible by all combinations of n fixed, distinct primes below a certain limit have to be added in counting the numbers divisible by at least 4 distinct primes below a certain limit.

			Let k be the number of values below 10^16 that are divisible by at least four distinct primes below 100. Let b(n) be the sum of all values below 10^16 that are divisible by all combinations of n fixed, distinct primes below 100. Then k = b(4) - 5*b(5) + 9*b(6) - 21*b(7) + 34*b(8) - 57*b(9) + 83*b(10) - 121*b(11) + 164*b(12) - 221*b(13).

Andrew Howroyd, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A002663 (number of times an element with n distinct prime factors will be counted, while taking the count for all the combinations of elements with some 4 fixed, distinct prime factors).

for(n=3,n=50,print((-1)^(n-3)*binomial(n,3)-1))

a(n) = (-1)^(n+1)*binomial(n,3) + 1.
|a(n+1)-a(n) | = A000330(n).
G.f. x*(5+6*x+4*x^2+x^3) / ( (x-1)*(1+x)^4 ). - R. J. Mathar, Sep 07 2012
a(n) = sum(k=1..n-3, (-1)^k*C(n+1,k)). - Mircea Merca, Feb 07 2013

cp's OEIS Frontend

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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A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

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A342379 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^3/6 - x^2/2 - x - 1).

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A347435 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6) ).

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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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A106471 A number triangle with duplicated columns of the form 2^n - Sum_{j=0..2k-1} C(n,j).

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A124326 T(n,m) = A007318(n,m) - A077028(n,m).

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A135552 Riordan array (1/((1-2x)(1-x)^2), -x/(1-x)^2).

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