OFFSET
0,4
COMMENTS
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Wikipedia, Permutation
FORMULA
MAPLE
h:= proc(n) option remember; `if`(n<1, 0, ilog2(n)+h(n-1)) end:
b:= proc(u, o) option remember; local n, g, l; n:= u+o;
if n=0 then 1
else g:= 2^ilog2(n); l:= min(g-1, n-g/2); expand(
add(x^(n-j)*add(binomial(j-1, i)*binomial(n-j, l-i)*
b(i, l-i)*b(j-1-i, n-l-j+i), i=0..min(j-1, l)), j=1..u)+
add(x^(j-1)*add(binomial(j-1, i)*binomial(n-j, l-i)*
b(l-i, i)*b(n-l-j+i, j-1-i), i=0..min(j-1, l)), j=1..o))
fi
end:
a:= n-> coeff(b(n, 0), x, iquo(h(n), 2)):
seq(a(n), n=0..25);
MATHEMATICA
h[n_] := h[n] = If[n < 1, 0, Length[IntegerDigits[n, 2]] - 1 + h[n - 1]];
b[u_, o_] := b[u, o] = Module[{n, g, l}, n = u + o; If[n == 0, 1,
g = 2^(Length[IntegerDigits[n, 2]] - 1); l = Min[g - 1, n - g/2];
Expand[Sum[x^(n - j)*Sum[Binomial[j - 1, i]*Binomial[n - j, l - i]*
b[i, l-i]*b[j-1-i, n-l-j+i], {i, 0, Min[j - 1, l]}], {j, 1, u}] +
Sum[x^(j - 1)*Sum[Binomial[j - 1, i]*Binomial[n - j, l - i]*
b[l-i, i]*b[n-l-j+i, j-1-i], {i, 0, Min[j - 1, l]}], {j, 1, o}]]]];
a[n_] := Coefficient[b[n, 0], x, Quotient[h[n], 2]];
a /@ Range[0, 25] (* Jean-François Alcover, Apr 23 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 14 2019
STATUS
approved