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A359856 Number of permutations of [1..n] which are indecomposable by direct and skew sums.

%I A359856 #14 Feb 10 2023 22:15:27
%S A359856 1,1,0,0,2,22,202,1854,17866,183806,2029850,24081006,306486314,
%T A359856 4175102110,60708557626,939518187726,15430666746826,268214861561726,
%U A359856 4921023843969242,95066628485598126,1929291834938927210,41042364285004263262,913409469123533445754,21227246586149632119438
%N A359856 Number of permutations of [1..n] which are indecomposable by direct and skew sums.
%H A359856 Wikipedia, <a href="https://en.wikipedia.org/wiki/Skew_and_direct_sums_of_permutations">Skew and direct sums of permutations</a>
%F A359856 G.f.: 2*(2 - 1/F(x)) - F(x) where F(x) = Sum_{k>=0} k!*x^k.
%F A359856 G.f.: S(F(x)) - 2*F(x)^2 - F(x) + x + 1 where S(x) is the g.f. of A111111 and F(x) = Sum_{k>=1} k!*x^k.
%F A359856 a(n) ~ n! * (1 - 4/n - 2/n^2 - 10/n^3 - 64/n^4 - 506/n^5 - 4762/n^6 - 51824/n^7 - 638678/n^8 - 8777898/n^9 - 132990772/n^10 - ...). - _Vaclav Kotesovec_, Jan 19 2023
%e A359856 The only permutations of [1..4] which are indecomposable by direct and skew sums are 2413 and 3142.
%t A359856 nmax = 20; CoefficientList[Series[2*(2 - 1/Sum[k!*x^k, {k, 0, nmax}]) - Sum[k!*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 19 2023 *)
%o A359856 (PARI) seq(n)={my(p=sum(k=0, n, k!*x^k, O(x*x^n))); Vec(2*(2 - 1/p) - p)} \\ _Andrew Howroyd_, Jan 16 2023
%Y A359856 Cf. A003319, A111111.
%K A359856 nonn
%O A359856 0,5
%A A359856 _Ludovic Schwob_, Jan 16 2023