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A229078 Number of ascending runs in {1,...,n}^n.

%I A229078 #23 Feb 16 2025 08:33:20
%S A229078 0,1,7,63,736,10625,182736,3647119,82837504,2109289329,59500000000,
%T A229078 1841557146671,62041198952448,2259914256880657,88499197217837056,
%U A229078 3707501605224609375,165444235911082541056,7834451891982365825441,392371124973096027488256
%N A229078 Number of ascending runs in {1,...,n}^n.
%H A229078 Alois P. Heinz, <a href="/A229078/b229078.txt">Table of n, a(n) for n = 0..200</a>
%H A229078 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%H A229078 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F A229078 a(n) = n^(n-1)*(n*(n+2)-1)/2 for n>0, a(0) = 0.
%F A229078 E.g.f.: 1/2*W(-x)*(W(-x)^3+W(-x)^2-W(-x)-2)/(1+W(-x))^3, W(x) Lambert's function (principal branch).
%F A229078 a(n) = A062023(n) + A066274(n) for n>0.
%e A229078 a(1) = 1: [1].
%e A229078 a(2) = 7 = 2+2+1+2: [1,1], [2,1], [1,2], [2,2].
%p A229078 a:= n-> `if`(n=0, 0, n^(n-1)*(n*(n+2)-1)/2):
%p A229078 seq(a(n), n=0..25);
%Y A229078 Main diagonal of A229079.
%Y A229078 Cf. A062023 (nondescending runs), A066274.
%K A229078 nonn
%O A229078 0,3
%A A229078 _Alois P. Heinz_, Sep 12 2013