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Number of compositions of n such that the smallest part has multiplicity three.
2

%I #14 Dec 06 2021 10:09:55

%S 1,0,4,5,14,24,59,108,213,419,808,1522,2872,5366,9960,18362,33660,

%T 61364,111375,201273,362225,649413,1160289,2066355,3668840,6495542,

%U 11469453,20201295,35496670,62233609,108878818,190103797,331292391,576296824,1000766991

%N Number of compositions of n such that the smallest part has multiplicity three.

%H Joerg Arndt and Alois P. Heinz, <a href="/A241863/b241863.txt">Table of n, a(n) for n = 3..1000</a>

%F a(n) ~ (13*sqrt(5)-29)/300 * n^3 * ((1+sqrt(5))/2)^n. - _Vaclav Kotesovec_, May 01 2014

%F Equivalently, a(n) ~ n^3 * phi^(n-7) / 150, where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021

%p b:= proc(n, s) option remember; `if`(n=0, 1,

%p `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))

%p end:

%p a:= proc(n) local k; k:= 3;

%p add((p->add(coeff(p, x, i)*binomial(i+k, k),

%p i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)

%p end:

%p seq(a(n), n=3..40);

%t b[n_, s_] := b[n, s] = If[n == 0, 1, If[n<s, 0, Expand[Sum[b[n-j, s]*x, {j, s, n}]]]]; a[n_] := With[{k=3}, Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i + k, k], {i, 0, Exponent[p, x]}]][b[n-j*k, j+1]], {j, 1, n/k}]]; Table[a[n], {n, 3, 40}] (* _Jean-François Alcover_, Feb 09 2015, after Maple *)

%Y Column k=3 of A238342.

%Y Cf. A001622.

%K nonn

%O 3,3

%A _Joerg Arndt_ and _Alois P. Heinz_, Apr 30 2014