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A241862 Number of compositions of n such that the smallest part has multiplicity two.

%I A241862 #15 Dec 06 2021 10:08:55
%S A241862 1,0,4,3,10,18,35,60,121,217,391,709,1281,2283,4061,7185,12680,22291,
%T A241862 39051,68191,118767,206331,357596,618345,1066983,1837513,3158685,
%U A241862 5420335,9286086,15884155,27130404,46274560,78822938,134095757,227853408,386721362,655639944
%N A241862 Number of compositions of n such that the smallest part has multiplicity two.
%H A241862 Joerg Arndt and Alois P. Heinz, <a href="/A241862/b241862.txt">Table of n, a(n) for n = 2..1000</a>
%F A241862 a(n) ~ (1/4 - 11/(20*sqrt(5))) * n^2 * ((1+sqrt(5))/2)^n. - _Vaclav Kotesovec_, May 01 2014
%F A241862 Equivalently, a(n) ~ n^2 * phi^(n-5) / (2 * 5^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021
%p A241862 b:= proc(n, s) option remember; `if`(n=0, 1,
%p A241862       `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
%p A241862     end:
%p A241862 a:= proc(n) local k; k:= 2;
%p A241862       add((p->add(coeff(p, x, i)*binomial(i+k, k),
%p A241862        i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
%p A241862     end:
%p A241862 seq(a(n), n=2..40);
%t A241862 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=2}, 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, 2, 40}] (* _Jean-François Alcover_, Nov 07 2014, after Maple *)
%Y A241862 Column k=2 of A238342.
%Y A241862 Cf. A001622.
%K A241862 nonn
%O A241862 2,3
%A A241862 _Joerg Arndt_ and _Alois P. Heinz_, Apr 30 2014