A324074 - OEIS

%I #13 Apr 23 2021 05:06:48

%S 0,0,1,6,48,360,2880,25200,262080,2903040,34473600,439084800,

%T 5987520000,87178291200,1351263513600,22230464256000,397533007872000,

%U 7469435990016000,147254595231744000,3041127510220800000,65688354220769280000,1481637322979573760000

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

%H Alois P. Heinz, <a href="/A324074/b324074.txt">Table of n, a(n) for n = 0..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Heap.html">Heap</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_heap">Binary heap</a>

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

%p b:= proc(u, o) option remember; local n, g, l; n:= u+o;

%p       if n=0 then 1

%p     else g:= 2^ilog2(n); l:= min(g-1, n-g/2); expand(

%p          add(x^(n-j)*add(binomial(j-1, i)*binomial(n-j, l-i)*

%p          b(i, l-i)*b(j-1-i, n-l-j+i), i=0..min(j-1, l)), j=1..u)+

%p          add(x^(j-1)*add(binomial(j-1, i)*binomial(n-j, l-i)*

%p          b(l-i, i)*b(n-l-j+i, j-1-i), i=0..min(j-1, l)), j=1..o))

%p       fi

%p     end:

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

%p seq(a(n), n=0..25);

%t b[u_, o_] := b[u, o] = Module[{n, g, l}, n = u + o; If[n == 0, 1,

%t      g = 2^(Length[IntegerDigits[n, 2]]-1); l = Min[g-1, n-g/2]; Expand[

%t      Sum[x^(n - j)*Sum[Binomial[j - 1, i]*Binomial[n - j, l - i]*

%t      b[i, l-i]*b[j-1-i, n-l-j+i], {i, 0, Min[j - 1, l]}], {j, 1, u}] +

%t      Sum[x^(j - 1)*Sum[Binomial[j - 1, i]*Binomial[n - j, l - i]*

%t      b[l-i, i]*b[n-l-j+i, j-1-i], {i, 0, Min[j - 1, l]}], {j, 1,o}]]]];

%t a[n_] := With[{p=b[n, 0]}, CoefficientList[p, x].Range[0, Exponent[p, x]]];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Apr 23 2021, after _Alois P. Heinz_ *)

%Y Cf. A000523, A061168, A306393.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Feb 14 2019