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 A302353 a(n) = Sum_{k=0..n} k^n*binomial(2*n-k,n). 1
 1, 1, 7, 69, 936, 16290, 345857, 8666413, 250355800, 8191830942, 299452606190, 12095028921250, 534924268768540, 25710497506696860, 1334410348734174285, 74379234152676275325, 4431350132232658244400, 281020603194039519937590, 18900157831016574533520330, 1343698678390575915132318870 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS FORMULA 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 EXAMPLE For n = 4 we have: ------------------------ 0   1    2    3    [4] ------------------------ 0,  1,  17,   98,  354,  ... A000538 (partial sums of fourth powers) 0,  1,  18,  116,  470,  ... A101089 (partial sums of A000538) 0,  1,  19,  135,  605,  ... A101090 (partial sums of A101089) 0,  1,  20,  155,  760,  ... A101091 (partial sums of A101090) 0,  1,  21,  176, [936], ... A254681 (partial sums of A101091) ------------------------ therefore a(4) = 936. MATHEMATICA Join[{1}, Table[Sum[k^n Binomial[2 n - k, n], {k, 0, n}], {n, 19}]] Table[SeriesCoefficient[HurwitzLerchPhi[x, -n, 0]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 19}] CROSSREFS Cf. A002054, A031971, A265612, A293550, A293574, A302352. Sequence in context: A265033 A226270 A121351 * A059321 A217400 A077683 Adjacent sequences:  A302350 A302351 A302352 * A302354 A302355 A302356 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Apr 06 2018 STATUS approved

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Last modified July 21 21:14 EDT 2019. Contains 325199 sequences. (Running on oeis4.)