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%I #11 Dec 12 2020 04:44:18
%S 47,544,4232,25100,136516,666800,3142884,14024256,61637303,262474700,
%T 1109010890,4603058016,19018730793,77751623552,317106002688,
%U 1284961711836,5199893190893,20961427995916,84431958561230,339292817869492,1362880886322817,5466605564267372
%N Number of partitions of n where each part i is marked with a word of length i over a quaternary alphabet whose letters appear in alphabetical order and all four letters occur at least once in the partition.
%H Alois P. Heinz, <a href="/A293368/b293368.txt">Table of n, a(n) for n = 4..1000</a>
%F a(n) ~ c * 4^n, where c = 4.90673361196637084263021203165784685586076564592828337755053385514766785... - _Vaclav Kotesovec_, Oct 11 2017
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
%p end:
%p a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(4):
%p seq(a(n), n=4..30);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];
%t a[n_] := With[{k = 4}, Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t a /@ Range[4, 30] (* _Jean-François Alcover_, Dec 12 2020, after _Alois P. Heinz_ *)
%Y Column k=4 of A261719.
%K nonn
%O 4,1
%A _Alois P. Heinz_, Oct 07 2017
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