A115567 -id:A115567 - cp's OEIS frontend

A116082 a(n) = C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

0, 1, 3, 7, 15, 31, 63, 127, 254, 501, 967, 1815, 3301, 5811, 9907, 16383, 26332, 41225, 63003, 94183, 137979, 198439, 280599, 390655, 536154, 726205, 971711, 1285623, 1683217, 2182395, 2804011, 3572223, 4514872, 5663889, 7055731, 8731847

Offset: 0

Jonathan Vos Post, Mar 13 2006

Number of compositions with at most three parts distinct from 1 and with a sum at most n. - Beimar Naranjo, Mar 12 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. See Table 2 on page 370. [_Parthasarathy Nambi_, Jun 18 2009]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000217, A000225, A004006, A055795, A057703, A115567, A116082. [Parthasarathy Nambi, Jun 18 2009]

[n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: n in [0..40]]; // Vincenzo Librandi, Jun 21 2011

a:=n->n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: seq(a(n),n=0..35); # Emeric Deutsch, Apr 14 2006
seq(sum(binomial(n,k),k=1..7),n=0..35); # Zerinvary Lajos, Dec 14 2007

Table[Total[Binomial[n,Range[7]]],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,1,3,7,15,31,63,127},41](* Harvey P. Dale, Aug 05 2011 *)

for(n=0,30, print1(n*(n^6 -14*n^5 +112*n^4 -350*n^3 +1099*n^2 +364*n +3828)/5040, ", ")) \\ G. C. Greubel, Nov 25 2017

a(n) = A000580(n) + A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.
a(n) = A000580(n) + A115567(n).
a(n) = n*(n^6 - 14*n^5 + 112*n^4 - 350*n^3 + 1099*n^2 + 364*n + 3828)/5040. - Emeric Deutsch, Apr 14 2006
G.f.: x*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)/(1-x)^8. - R. J. Mathar, Jun 20 2011
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31, a(6)=63, a(7)=127. - Harvey P. Dale, Aug 05 2011

A131251 A000012 * A052509.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 7, 3, 1, 6, 15, 14, 7, 3, 1, 7, 21, 25, 15, 7, 3, 1, 8, 28, 41, 30, 15, 7, 3, 1, 9, 36, 63, 56, 31, 15, 7, 3, 1, 10, 45, 92, 98, 62, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A001924: (1, 3, 7, 14, 26, 46, 79, ...). A131252 = A052509 * A000012.
From Clark Kimberling, Feb 07 2011: (Start)
When formatted as a rectangle R with northwest corner
  1, 2, 3,  4,  5,  6, ...
  1, 3, 6, 10, 15, 21, ...
  1, 3, 7, 14, 25, 41, ...
  1, 3, 7, 15, 30, 56, ...
  1, 3, 7, 15, 31, 62, ...
  ...
the following properties hold:
R is the accumulation array of the transpose of A052553 (a version of Pascal's triangle); see A144112 for the definition of accumulation array.
row 1: A000027
row 2: A000217
row 3: A004006
row 4: A055795
row 5: A057703
row 6: A115567
limiting row:  A000225
antidiagonal sums: A001924.
(End)

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  3,  1;
  5, 10,  7,  3,  1;
  6, 15, 14,  7,  3,  1;
  7, 21, 25, 15,  7,  3,  1;
  ...

Cf. A052509, A000012, A001924, A131252.

A000012 * A052509 as infinite lower triangular matrices.

cp's OEIS Frontend

A116082 a(n) = C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

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