OFFSET
0,2
COMMENTS
a(n-1) is the n-th antidiagonal sum of the convolution array A213835. - Clark Kimberling, Jul 04 2012
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = binomial(n+3,3)*(2*n+1) = (n+1)*(n+2)*(n+3)*(2*n+1)/6.
G.f.: (1+7*x)/(1-x)^5.
a(n) = A080851(8,n). - R. J. Mathar, Jul 28 2016
E.g.f.: (6 + 66*x + 81*x^2 + 25*x^3 + 2*x^4)*exp(x)/6. - G. C. Greubel, Aug 30 2019
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = (32*log(2) - 11)/10.
Sum_{n>=0} (-1)^n/a(n) = (8*Pi - 56*log(2) + 23)/10. (End)
MAPLE
seq((2*n+1)*binomial(n+3, 3), n=0..40); # G. C. Greubel, Aug 30 2019
MATHEMATICA
Table[(2*n+1)*Binomial[n+3, 3], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011, modified by G. C. Greubel, Aug 30 2019 *)
PROG
A001107:=func<n | n*(4*n-3)>; [&+[(n-i+1)*A001107(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012
(Magma) [(2*n+1)*Binomial(n+3, 3): n in [0..40]]; // G. C. Greubel, Aug 30 2019
(PARI) vector(40, n, (2*n-1)*binomial(n+2, 3)) \\ G. C. Greubel, Aug 30 2019
(Sage) [(2*n+1)*binomial(n+3, 3) for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> (2*n+1)*Binomial(n+3, 3)) # G. C. Greubel, Aug 30 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Barry E. Williams, Dec 11 1999
STATUS
approved