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Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order such that sorts of adjacent parts are different.
3

%I #10 Feb 05 2017 04:37:49

%S 1,1,2,4,10,27,87,312,1269,5703,28082,149643,855938,5217753,33712046,

%T 229799508,1646314498,12355371024,96861186897,791258791159,

%U 6720627161903,59234364141343,540812222291531,5106663817387466,49798678281320763,500857393909589995

%N Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order such that sorts of adjacent parts are different.

%H Alois P. Heinz, <a href="/A262496/b262496.txt">Table of n, a(n) for n = 0..400</a>

%e a(3) = 4: 3a, 2a1b, 1a1b1a, 1a1b1c (in this example the sorts are labeled a, b, c).

%p b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),

%p b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))

%p end:

%p A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):

%p T:= (n, k)-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):

%p a:= n-> add(T(n, k), k=0..n):

%p seq(a(n), n=0..30);

%t b[n_, i_, k_] := b[n, i, k] = If[n==0 || i==1, k^(n-1), b[n, i-1, k] + If[i > n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n==0, 1, If[k<2, k, k*b[n, n, k - 1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 05 2017, translated from Maple *)

%Y Row sums of A262495.

%Y Cf. A258466.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 24 2015