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 A302353 a(n) = Sum_{k=0..n} k^n*binomial(2*n-k,n). 1

%I

%S 1,1,7,69,936,16290,345857,8666413,250355800,8191830942,299452606190,

%T 12095028921250,534924268768540,25710497506696860,1334410348734174285,

%U 74379234152676275325,4431350132232658244400,281020603194039519937590,18900157831016574533520330,1343698678390575915132318870

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

%C 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 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 For n = 4 we have:

%e ------------------------

%e 0 1 2 3 [4]

%e ------------------------

%e 0, 1, 17, 98, 354, ... A000538 (partial sums of fourth powers)

%e 0, 1, 18, 116, 470, ... A101089 (partial sums of A000538)

%e 0, 1, 19, 135, 605, ... A101090 (partial sums of A101089)

%e 0, 1, 20, 155, 760, ... A101091 (partial sums of A101090)

%e 0, 1, 21, 176, [936], ... A254681 (partial sums of A101091)

%e ------------------------

%e therefore a(4) = 936.

%t Join[{1}, Table[Sum[k^n Binomial[2 n - k, n], {k, 0, n}], {n, 19}]]

%t Table[SeriesCoefficient[HurwitzLerchPhi[x, -n, 0]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 19}]

%Y Cf. A002054, A031971, A265612, A293550, A293574, A302352.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 06 2018

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Last modified August 20 07:48 EDT 2019. Contains 326143 sequences. (Running on oeis4.)