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A320546
Number of partitions of n into parts of exactly four sorts which are introduced in ascending order such that sorts of adjacent parts are different.
3
1, 7, 33, 130, 464, 1558, 5039, 15886, 49282, 151165, 460352, 1394863, 4212752, 12694566, 38197710, 114820403, 344919283, 1035670246, 3108844526, 9330186438, 27997888759, 84008273161, 252054096569, 756220672185, 2268778953179, 6806570182252, 20420177671614
OFFSET
4,2
LINKS
FORMULA
a(n) ~ 3^(n-1) / (3! * QPochhammer[1/3]). - Vaclav Kotesovec, Oct 25 2018
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))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(4):
seq(a(n), n=4..40);
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]]];
a[n_] := With[{k = 4}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
a /@ Range[4, 40] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)
CROSSREFS
Column k=4 of A262495.
Cf. A258459.
Sequence in context: A229515 A320907 A258458 * A066810 A262600 A034577
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 15 2018
STATUS
approved