A035038 -id:A035038 - 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).

A055250 Seventh column of triangle A055249.

Original entry on oeis.org

1, 9, 47, 187, 630, 1898, 5282, 13866, 34831, 84575, 199977, 462973, 1053804, 2365704, 5250660, 11543700, 25177005, 54539205, 117456115, 251676495, 536892146, 1140875254, 2415947382, 5100306062, 10737455195, 22548620283

Offset: 0

Wolfdieter Lang, May 26 2000

Cf. A055249, A035038, partial sums of A034009.

a:= n-> (Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [9,-34,70,-85,61,-24,4][i] else 0 fi)^(n))[1,1]: seq(a(n), n=0..25); # Alois P. Heinz, Aug 05 2008

Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 5, k + 5], {k, 0, n}], {n, 1, 26}] (* Zerinvary Lajos, Jul 08 2009 *)

G.f.: 1/(((1-2*x)^2)*(1-x)^5).
a(n) = A055249(n+6, 6).
For n >= 1, a(n) = A035038(n+6) + Sum_{j=0..n-1} a(j).
a(n) = Sum_{k=0..n+5} Sum_{i=0..n+5} (i-k) * C(n-k+5,i+3). - Wesley Ivan Hurt, Sep 19 2017

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).

A084637 Binomial transform of (1,0,1,0,1,0,1,1,1,1,1,...).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 65, 136, 293, 642, 1410, 3072, 6606, 14004, 29295, 60592, 124187, 252742, 511672, 1031912, 2075452, 4166408, 8353165, 16732664, 33498977, 67040458, 134134046, 268333872, 536748474, 1073595228, 2147309211, 4294760928, 8589691767

Offset: 0

Paul Barry, Jun 06 2003

The sequence starting 1,2,4,... is the binomial transform of (1, 1, 1, 1, 1, 1, 2, 2, 2, ...) with A035038(n) = Sum_{k=0..5} C(n,k) + 2*Sum_{k=6..n} C(n,k) = 2^n - (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. This gives the partial sums of A084636.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-27,50,-55,36,-13,2).

Cf. A000225, A000325, A035038, A084634, A084635, A084636.

[2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120: n in [0..50]]; // G. C. Greubel, Mar 19 2023

Table[2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120, {n,0,50}] (* G. C. Greubel, Mar 19 2023 *)

Vec((1-7*x+21*x^2-35*x^3+35*x^4-21*x^5+7*x^6)/((1-x)^6*(1-2*x)) + O(x^50)) \\ Colin Barker, Mar 17 2016

[2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120 for n in range(51)] # G. C. Greubel, Mar 19 2023

a(n) = Sum_{k=0..2} C(n, 2*k) + Sum_{k=6..n} C(n, k).
a(n) = 2^n - n*(n^4 - 10*n^3 + 55*n^2 - 110*n + 184)/120.
G.f.: (1-7*x+21*x^2-35*x^3+35*x^4-21*x^5+7*x^6) / ((1-x)^6*(1-2*x)). - Colin Barker, Mar 17 2016

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)).

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

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

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

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A055250 Seventh column of triangle A055249.

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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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A084637 Binomial transform of (1,0,1,0,1,0,1,1,1,1,1,...).

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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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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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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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