A262496 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.

1, 1, 2, 4, 10, 27, 87, 312, 1269, 5703, 28082, 149643, 855938, 5217753, 33712046, 229799508, 1646314498, 12355371024, 96861186897, 791258791159, 6720627161903, 59234364141343, 540812222291531, 5106663817387466, 49798678281320763, 500857393909589995

Offset: 0

Author

Alois P. Heinz, Sep 24 2015

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Row sums of A262495.

Cf. A258466.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
T:= (n, k)->  add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..30);

Mathematica

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 *)