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A053729 Self-convolution of 1,4,27,256,3125,46656,... (cf. A000312).

%I A053729 #26 Jul 02 2025 16:01:59
%S A053729 1,8,70,728,9027,132136,2254620,44262200,987183525,24718587592,
%T A053729 687457908306,21034757596184,702270963692039,25400848001674856,
%U A053729 989240042333246072,41263578858484555512,1835070614332428285513
%N A053729 Self-convolution of 1,4,27,256,3125,46656,... (cf. A000312).
%H A053729 John Tyler Rascoe, <a href="/A053729/b053729.txt">Table of n, a(n) for n = 1..100</a>
%F A053729 a(n) = Sum_{k=1..n} k^k * (n+1-k)^(n+1-k).
%F A053729 a(n) ~ 2 * n^n. - _Vaclav Kotesovec_, Mar 10 2018
%e A053729 a(4) = 1^1 *4^4 +2^2 *3^3 +3^3 *2^2 +4^4 *1^1 = 1*256 +4*27 +27*4 +256*1 = 728.
%t A053729 nn=20;f[x_]=Sum[n^n x^n,{n,1,nn}];CoefficientList[Series[f[x]^2/x^2,{x,0,nn}],x] (* _Geoffrey Critzer_, Nov 05 2013 *)
%t A053729 Table[Sum[k^k*(n+1-k)^(n+1-k), {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Mar 10 2018 *)
%o A053729 (Python)
%o A053729 def A053729(n): return sum((k**k)*(n+1-k)**(n+1-k) for k in range(1,n+1)) # _John Tyler Rascoe_, Aug 23 2024
%Y A053729 Cf. A000312, A100262.
%K A053729 nonn,easy
%O A053729 1,2
%A A053729 _Leroy Quet_, Feb 11 2000
%E A053729 More terms from _James Sellers_, Feb 22 2000