A324074 Total number of distorted ancestor-successor pairs in all defective (binary) heaps on n elements.

0, 0, 1, 6, 48, 360, 2880, 25200, 262080, 2903040, 34473600, 439084800, 5987520000, 87178291200, 1351263513600, 22230464256000, 397533007872000, 7469435990016000, 147254595231744000, 3041127510220800000, 65688354220769280000, 1481637322979573760000

Offset: 0

Author

Alois P. Heinz, Feb 14 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Eric Weisstein's World of Mathematics, Heap
• Wikipedia, Binary heap

Crossrefs

Cf. A000523, A061168, A306393.

Programs

Maple

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-> (p-> add(coeff(p, x, i)*i, i=0..degree(p)))(b(n, 0)):
seq(a(n), n=0..25);

Mathematica

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_] := With[{p=b[n, 0]}, CoefficientList[p, x].Range[0, Exponent[p, x]]];
a /@ Range[0, 25] (* Jean-François Alcover, Apr 23 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..A061168(n)} k * A306393(n,k).