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

0,2

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

References

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 216.

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).

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)

E.g.f.: exp(x)*(240 + 1920*x + 3000*x^2 + 1600*x^3 + 350*x^4 + 32*x^5 + x^6)/240. - Stefano Spezia, Oct 30 2025

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000217, A027801, A051836.

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

Sequence in context: A000971 A061927 A292481 * A180670 A268262 A293101

Adjacent sequences: A051920 A051921 A051922 * A051924 A051925 A051926

Keyword

nonn,easy

Author

Barry E. Williams, Dec 19 1999

Status

approved