cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A324069 Number of defective (binary) heaps on n elements where eight ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 20, 480, 4830, 40320, 346080, 3014400, 28036800, 271180800, 2723635200, 28751923200, 273405132000, 2754444492800, 29222409216000, 335670386688000, 3786723502848000, 45941770321920000, 580488335032320000, 8000481890598912000
Offset: 0

Views

Author

Alois P. Heinz, Feb 13 2019

Keywords

Comments

Or number of permutations p of [n] having exactly eight pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Links

Crossrefs

Column k=8 of A306393.
Cf. A056971.

A324070 Number of defective (binary) heaps on n elements where nine ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 240, 4200, 39424, 359520, 3340800, 32630400, 331499520, 3503385600, 38504294400, 386486100000, 4064835174400, 44847772569600, 530121646080000, 6258102529536000, 78618109870080000, 1027628834918400000, 14504975258222592000
Offset: 0

Views

Author

Alois P. Heinz, Feb 13 2019

Keywords

Comments

Or number of permutations p of [n] having exactly nine pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Links

Crossrefs

Column k=9 of A306393.
Cf. A056971.

A324071 Number of defective (binary) heaps on n elements where ten ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 80, 3150, 36736, 359520, 3532800, 36115200, 383708160, 4247443200, 48673996800, 515675160000, 5654852403200, 64785924403200, 788119068672000, 9695238119424000, 125961866477568000, 1700829800017920000, 24580421999198208000
Offset: 0

Views

Author

Alois P. Heinz, Feb 13 2019

Keywords

Comments

Or number of permutations p of [n] having exactly ten pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Links

Crossrefs

Column k=10 of A306393.
Cf. A056971.

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

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Feb 14 2019

Keywords

Links

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).

A324075 Number of defective (binary) heaps on n elements having one half of their ancestor-successor pairs (rounded down) distorted.

Original entry on oeis.org

1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 359520, 3590400, 39362400, 472919040, 6133670400, 85948262400, 1284106824000, 20434058444800, 345796766515200, 6188467544064000, 117398964114432000, 2341018467532800000, 49035684501872640000, 1074839883779211264000
Offset: 0

Views

Author

Alois P. Heinz, Feb 14 2019

Keywords

Comments

Number of permutations p of [n] having exactly floor(A061168(n)/2) pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).
Central terms (also maxima) of rows of A306393.

Links

Crossrefs

Cf. A000142, A000523, A061168, A306393.

Programs

  • 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 *)

Formula

a(n) = A306393(floor(A061168(n)/2)).
a(n) <= (n-1)! for n >= 1 with equality only for n <= 9.
Previous Showing 11-15 of 15 results.

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