A051923 Partial sums of A051836.

1, 9, 42, 140, 378, 882, 1848, 3564, 6435, 11011, 18018, 28392, 43316, 64260, 93024, 131784, 183141, 250173, 336490, 446292, 584430, 756470, 968760, 1228500, 1543815, 1923831, 2378754, 2919952, 3560040, 4312968, 5194112, 6220368, 7410249, 8783985, 10363626

Offset: 0

Author

Barry E. Williams, Dec 19 1999

Keywords

• nonn
• easy

Comments

If Y is a 3-subset of an n-set X then, for n >= 8, a(n-8) is the number of 8-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007

a(n) is the n-th antidiagonal sum of the convolution array A213551. - Clark Kimberling, Jun 17 2012

Examples

			From the third formula: a(4) = 15+60+108+120+75 = 378. - _Bruno Berselli_, Sep 04 2013
		

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000217, A027801, A051836.

Cf. A093560 ((3, 1) Pascal, column m=6).

Programs

Magma

[Binomial(n+5, 5)*(n+2)/2: n in [0..40]]; // Vincenzo Librandi, Dec 27 2018

Mathematica

CoefficientList[Series[(1 + 2 x)/(1 - x)^7, {x, 0, 25}], x]  (* Harvey P. Dale, Mar 13 2011 *)
Nest[Accumulate,Range[1,120,3],5] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
Table[Binomial[n + 5, 5] (n + 2) / 2, {n, 0, 35}] (* Vincenzo Librandi, Dec 27 2018 *)

Formula

a(n) = binomial(n+5, 5)*(n+2)/2.

G.f.: (1+2*x)/(1-x)^7.

a(n) = Sum_{k=1..n+1} k*A000217(k)*A000217(n-k+2). - Bruno Berselli, Sep 04 2013

From Amiram Eldar, Jan 28 2022: (Start)

Sum_{n>=0} 1/a(n) = 1205/18 - 20*Pi^2/3.

Sum_{n>=0} (-1)^n/a(n) = 10*Pi^2/3 - 320*log(2)/3 + 755/18. (End)