%I #36 Feb 22 2025 09:57:06
%S 3,192,3000,24000,128625,526848,1778112,5184000,13476375,31944000,
%T 70180968,144685632,282589125,526848000,943296000,1630015488,
%U 2729559627,4444632000,7057911000,10956792000,16663911033,24874409472,36501000000,52728000000,75075609375
%N a(n) = n^3*(n+1)^3*(n+2)^3/72.
%D Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 162.
%H Paolo Xausa, <a href="/A202109/b202109.txt">Table of n, a(n) for n = 1..10000</a>
%H Pedro A. Piza, <a href="http://www.jstor.org/stable/3029445">Powers of sums and sums of powers</a>, Math. Mag. 25 (3) (1952) 137.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F a(n) = 3*(Sum_{k=1..n} k*(k+1)/2)^3.
%F a(n) = 3*A000292(n)^3.
%F a(n) = Sum_{k=1..n} A000217(k)^3+2*A000217(k)^4.
%F G.f.: 3*x*(1+54*x+405*x^2+760*x^3+405*x^4+54*x^5+x^6) / (x-1)^10. - _R. J. Mathar_, Dec 13 2011
%F From _Amiram Eldar_, Apr 09 2024: (Start)
%F Sum_{n>=1} 1/a(n) = 261/4 - 54*zeta(3).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 135*zeta(3)/2 + 432*log(2) - 1521/4. (End)
%F E.g.f.: exp(x)*x*(216 + 6696*x + 29196*x^2 + 39420*x^3 + 22032*x^4 + 5796*x^5 + 747*x^6 + 45*x^7 + x^8)/72. - _Stefano Spezia_, Feb 21 2025
%t Array[#^3*(#+1)^3*(#+2)^3/72 &, 50] (* _Paolo Xausa_, Apr 07 2024 *)
%Y Cf. A000217, A000292, A002117.
%K nonn,easy,changed
%O 1,1
%A _Martin Renner_, Dec 11 2011