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A306948 Expansion of e.g.f. (1 + x)*log(1 + x)*exp(x). 0
0, 1, 3, 5, 8, 9, 19, -15, 216, -1407, 11803, -108483, 1106192, -12363703, 150381243, -1977666743, 27965386320, -423158076351, 6822782712723, -116781368777867, 2114916140765496, -40404117909336247, 812091479233464131, -17130720178674680031, 378423227774537955688 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(k)*k.
a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)*(n - k + 1)*(k - 1)!.
a(n) ~ exp(-1) * (-1)^n * n! / n^2. - Vaclav Kotesovec, Mar 18 2019
Conjecture: D-finite with recurrence a(n) +(n-5)*a(n-1) +(-3*n+10)*a(n-2) +3*(n-3)*a(n-3) +(-n+3)*a(n-4)=0. - R. J. Mathar, Aug 20 2021
MAPLE
a:=series((1 + x)*log(1 + x)*exp(x), x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019
MATHEMATICA
nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] BellB[k] k, {k, 0, n}], {n, 0, 24}]
Table[Sum[(-1)^(k - 1) Binomial[n, k] (n - k + 1) (k - 1)!, {k, 1, n}], {n, 0, 24}]
CROSSREFS
Sequence in context: A343114 A349679 A355761 * A067241 A360030 A120943
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Mar 17 2019
STATUS
approved

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Last modified July 16 12:01 EDT 2024. Contains 374348 sequences. (Running on oeis4.)