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A089974
(n+1)-st term of the n-th binomial transform of this sequence equals 1 for all n>=0.
0
1, 0, -3, 1, 17, -49, -71, 1079, -3263, -10529, 156401, -624601, -1858319, 42109055, -249633047, -187690049, 17346988289, -159631097281, 375894263521, 8905140874583, -137481700309199, 830604602372959, 3549053339380121, -139582700722475473, 1526848956867592513
OFFSET
0,3
COMMENTS
Equals row sums of triangle A089962, which is the matrix inverse of triangle A071207.
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n, k)*n*k^(n-k-1).
Sum_{k=0..n} C(n, k)*n^(n-k)*a(k) = 1.
O.g.f.: A(x) = (1-x)*sum_{n>=0} x^n/(1+n*x)^(n+2).
E.g.f.: (1-x)*exp(x*exp(-x)).
EXAMPLE
Note the diagonal of 1's in the array of iterated binomial transforms:
[1, 0, -3, 1, 17, -49, -71, 1079,..]
[1, 1, -2, -7, 4, 17, -134, 121,..]
[1, 2, 1, -9, -31, -47, -135, -599,..]
[1, 3, 6, 1, -52, -271, -1034, -3895,..]
[1, 4, 13, 29, 1, -445, -3311, -18461,..]
[1, 5, 22, 81, 212, 1, -5166, -49631,..]
[1, 6, 33, 163, 689, 2117, 1, -75019,..]
[1, 7, 46, 281, 1564, 7553, 26830, 1,..]
MATHEMATICA
Join[{1, 0}, Table[Sum[(-1)^(n - k)*Binomial[n, k]*n*k^(n - k - 1), {k, 0, n} ], {n, 2, 25}]] (* Vaclav Kotesovec, Jun 20 2022 *)
PROG
(PARI) a(n)=if(n<0, 0, if(n==0, 1, sum(k=0, n, (-1)^(n-k)*binomial(n, k)*n*k^(n-k-1) )))
CROSSREFS
Sequence in context: A060281 A350078 A151918 * A346039 A143849 A370233
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 18 2003
EXTENSIONS
a(19) and a(20) concatenated by Georg Fischer, Jun 20 2022
STATUS
approved