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A051923
Partial sums of A051836.
9
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.
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. A093560 ((3, 1) Pascal, column m=6).
Sequence in context: A000971 A061927 A292481 * A180670 A268262 A293101
KEYWORD
nonn,easy
AUTHOR
Barry E. Williams, Dec 19 1999
STATUS
approved