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A302353 a(n) = Sum_{k=0..n} k^n*binomial(2*n-k,n).

%I A302353 #8 Apr 08 2018 05:36:11
%S A302353 1,1,7,69,936,16290,345857,8666413,250355800,8191830942,299452606190,
%T A302353 12095028921250,534924268768540,25710497506696860,1334410348734174285,
%U A302353 74379234152676275325,4431350132232658244400,281020603194039519937590,18900157831016574533520330,1343698678390575915132318870
%N A302353 a(n) = Sum_{k=0..n} k^n*binomial(2*n-k,n).
%C A302353 a(n) is the n-th term of the main diagonal of iterated partial sums array of n-th powers (starting with the first partial sums).
%F A302353 a(n) ~ c * (r * (2-r)^(2-r) / (1-r)^(1-r))^n * n^n, where r = 0.69176629470097668698335106516328398961170464277337300459988208658267146... is the root of the equation (2-r) = (1-r) * exp(1/r) and c = 0.96374921279011282619632879505754646526289414675402231447188230355850496... - _Vaclav Kotesovec_, Apr 08 2018
%e A302353 For n = 4 we have:
%e A302353 ------------------------
%e A302353 0   1    2    3    [4]
%e A302353 ------------------------
%e A302353 0,  1,  17,   98,  354,  ... A000538 (partial sums of fourth powers)
%e A302353 0,  1,  18,  116,  470,  ... A101089 (partial sums of A000538)
%e A302353 0,  1,  19,  135,  605,  ... A101090 (partial sums of A101089)
%e A302353 0,  1,  20,  155,  760,  ... A101091 (partial sums of A101090)
%e A302353 0,  1,  21,  176, [936], ... A254681 (partial sums of A101091)
%e A302353 ------------------------
%e A302353 therefore a(4) = 936.
%t A302353 Join[{1}, Table[Sum[k^n Binomial[2 n - k, n], {k, 0, n}], {n, 19}]]
%t A302353 Table[SeriesCoefficient[HurwitzLerchPhi[x, -n, 0]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 19}]
%Y A302353 Cf. A002054, A031971, A265612, A293550, A293574, A302352.
%K A302353 nonn
%O A302353 0,3
%A A302353 _Ilya Gutkovskiy_, Apr 06 2018