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

1, 1, 7, 69, 936, 16290, 345857, 8666413, 250355800, 8191830942, 299452606190, 12095028921250, 534924268768540, 25710497506696860, 1334410348734174285, 74379234152676275325, 4431350132232658244400, 281020603194039519937590, 18900157831016574533520330, 1343698678390575915132318870

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

Table of n, a(n) for n=0..19.

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