A035042 -id:A035042 - 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).

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

cp's OEIS Frontend

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

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