A067699 -id:A067699 - cp's OEIS frontend

A063090 a(n)/(n*n!) is the average number of comparisons needed to find a node in a binary search tree containing n nodes inserted in a random order.

1, 6, 34, 212, 1488, 11736, 103248, 1004832, 10733760, 124966080, 1575797760, 21403457280, 311623441920, 4842481190400, 80007869491200, 1400671686758400, 25902542427955200, 504597366114508800, 10328835149402112000

Offset: 1

Rob Arthan, Aug 06 2001

a(n) is the sum over all permutations, p, of {1, ..., n} of the number of comparisons required to find all the entries in the tree formed when the order of insertion is p(1), p(2), ... p(n). To derive the formula given, first group the trees according to the value of k = p(1). For a given k, p determines a permutation of {1, ..., k-1} that gives the structure of the left subtree. By symmetry, the contribution of the right subtrees will be the same as the left subtrees. Now count and simplify.

a(n) mod n is n-2 or 0 depending on whether n is prime or not. - Gary Detlefs, May 28 2012

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 427, C(n).

T. D. Noe, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Quicksort.

Cf. A000108, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971.

[Factorial(n)*((2*n+2)*HarmonicNumber(n) - 3*n): n in [1..30]]; // G. C. Greubel, Sep 01 2018

A[1]:= 1:
for n from 2 to 30 do A[n]:= (2*n-1)*(n-1)!+(n+1)*A[n-1] od:
seq(A[n],n=1..30); # Robert Israel, Sep 21 2014

a[n_] := n!*((2*n+2)*HarmonicNumber[n] - 3*n); Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 20 2012, after Gary Detlefs *)

{h(n) = sum(k=1,n, 1/k)};
for(n=1,30, print1(n!*(2*(n+1)*h(n) - 3*n), ", ")) \\ G. C. Greubel, Sep 01 2018

a(1) = 1, a(n) = n*n! + 2 * Sum_{k=1}^{n-1} (n-1)!/k! * a(k).
a(n) = (2*n - 1)*(n - 1)! + (n + 1)*a(n-1).
E.g.f.: -(x+2*log(1-x))/(1-x)^2. - Vladeta Jovovic, Sep 15 2003
a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A000337(k). - Vladeta Jovovic, Jul 06 2004
a(n) = 2*(n+1)*abs(Stirling1(n+1, 2))-3*n*n!. - Vladeta Jovovic, Jul 06 2004
a(n) = n!*((2*n+2)*h(n) - 3*n), where h(n) is the n-th harmonic number. - Gary Detlefs, May 28 2012
a(n) = A288964(n) + n!*n (because this sequence and A288964 have the same definition related to quicksort but under slightly different assumptions). - Petros Hadjicostas, May 03 2020

More terms from Vladeta Jovovic, Aug 08 2001

Missing brackets in the formula in the name inserted by Rob Arthan, Sep 21 2014

A330852 Numerator of the rational number A(n) that appears in the formula for the n-th cumulant k(n) = (-1)^n2^n(A(n) - (n - 1)!*zeta(n)) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, with A(0) = 1 and A(1) = 0.

Original entry on oeis.org

1, 0, 7, 19, 937, 85981, 21096517, 7527245453, 19281922400989, 7183745930973701, 3616944955616896387, 273304346447259998403709, 76372354431694636659849988531, 25401366514997931592208126670898607, 110490677504100075544188675746430710672527, 37160853195949529205295416197788818165029489819

Offset: 0

Petros Hadjicostas, Apr 28 2020

Hennequin conjectured his cumulant formula in his 1989 paper and proved it in his 1991 thesis.
First he calculates the numbers (B(n): n >= 0), with B(0) = 1 and B(0) = 0, given for p >= 0 by the recurrence
Sum_{r=0..p} Stirling1(p+2, r+1)*B(p-r)/(p-r)! + Sum_{r=0..p} F(r)*F(p-r) = 0, where F(r) = Sum_{i=0..r} Stirling1(r+1,i+1)*G(r-i) and G(k) = Sum_{a=0..k} (-1)^a*B(k-a)/(a!*(k-a)!*2^a).
Then A(n) = L_n(B(1),...,B(n)), where L_n(x_1,...,x_n) are the logarithmic polynomials of Bell.
Hoffman and Kuba (2019, 2020) gave an alternative proof of Hennequin's cumulant formula and gave an alternative calculation for the constants (-2)^n*A(n), which they denote by a_n. See also Finch (2020).
The Maple program below is based on Tan and Hadjicostas (1993), where the numbers (A(n): n >= 0) are also tabulated.
The rest of the references give the theory of the limiting distribution of the number of comparisons in quicksort (and for that reason we omit any discussion of that topic).

			The first few fractions A(n) are
1, 0, 7/4, 19/8, 937/144, 85981/3456, 21096517/172800, 7527245453/10368000, 19281922400989/3810240000, 7183745930973701/177811200000, ...
The first few fractions (-2)^n*A(n) (= a_n in Hoffman and Kuba and in Finch) are
1, 0, 7, -19, 937/9, -85981/108, 21096517/2700, -7527245453/81000, 19281922400989/14883750, -7183745930973701/347287500, ...

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991), p. 83.

Petros Hadjicostas, Table of n, a(n) for n = 0..30
S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the first eight moments for the number of comparisons in quicksort from which Hennequin's first eight asymptotic cumulants can be derived.]
James A. Fill and Svante Janson, Smoothness and decay properties of the limiting Quicksort density function, In: D. Gardy and A. Mokkadem (eds), Mathematics and Computer Science, Trends in Mathematics, Birkhäuser, Basel, 2000, pp. 53-64.
James A. Fill and Svante Janson, Quicksort asymptotics, Journal of Algorithms, 44(1) (2002), 4-28.
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives the constants a_s = (-2)^s*A(s) for s >= 2.]
P. Hennequin, Combinatorial analysis of the quicksort algorithm, Informatique théoretique et applications, 23(3) (1989), 317-333.
M. E. Hoffman and M. Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020; see Section 5.2. [They study the constants a_s = (-2)^s*A(s) for s >= 2.]
Mireille Régnier, A limiting distribution for quicksort, Informatique théorique et applications, 23(3) (1989), 335-343.
Uwe Rösler, A limit theorem for quicksort, Informatique théorique et applications, 25(1) (1991), 85-100.
Kok Hooi Tan and Petros Hadjicostas, Density and generating functions of a limiting distribution in quicksort, Technical Report #568, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA, 1993; see p. 10.
Kok Hooi Tan and Petros Hadjicostas, Some properties of a limiting distribution in Quicksort, Statistics and Probability Letters, 25(1), 1995, 87-94.
Vytas Zacharovas, On the exponential decay of the characteristic function of the quicksort distribution, arXiv:1605.04018 [math.CO], 2016.

Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A330860 (denominators).

# Produces the sequence (B(n): n >= 0)
B := proc(m) option remember: local v, g, f, b:
if m = 0 then v := 1: end if: if m = 1 then v := 0: end if:
if 2 <= m then
g := proc(k) add((-1)^a*B(k - a)/(a!*(k - a)!*2^a), a = 0 .. k): end proc:
f := proc(r) add(Stirling1(r + 1, i + 1)*g(r - i), i = 0 .. r): end proc:
b := proc(p) (-1)^p*(add(Stirling1(p + 2, r + 1)*B(p - r)/(p - r)!, r = 1 .. p) + add(f(rr)*f(p - rr), rr = 1 .. p - 1) + 2*(-1)^p*p!*add((-1)^a*B(p - a)/(a!*(p - a)!*2^a), a = 1 .. p) + 2*add(Stirling1(p + 1, i + 1)*g(p - i), i = 1 .. p))/(p - 1): end proc:
v := simplify(b(m)): end if: v: end proc:
# Produces the sequence (A(n): n >= 0)
A := proc(m) option remember: local v:
if m = 0 then v := 1: end if: if m = 1 then v := 0: end if:
if 2 <= m then v := -(m - 1)!*add(A(k + 1)*B(m - 1 - k)/(k!*(m - 1 - k)!), k = 0 .. m - 2) + B(m): end if: v: end proc:
# Produces the sequence of numerators of the A(n)'s
seq(numer(A(n)), n = 0 .. 15);

B[m_] := B[m] = Module[{v, g, f, b}, If[m == 0, v = 1]; If[m == 1, v = 0]; If[2 <= m, g[k_] := Sum[(-1)^a*B[k - a]/(a!*(k - a)!*2^a), {a, 0, k}]; f[r_] := Sum[StirlingS1[r + 1, i + 1]*g[r - i], {i, 0, r}]; b[p_] := (-1)^p*(Sum[StirlingS1[p + 2, r + 1]*B[p - r]/(p - r)!, {r, 1, p}] + Sum[f[rr]*f[p - rr], {rr, 1, p - 1}] + 2*(-1)^p*p!*Sum[(-1)^a*B[p - a]/(a!*(p - a)!*2^a), {a, 1, p}] + 2*Sum[StirlingS1[p + 1, i + 1]*g[p - i], {i, 1, p}])/(p - 1); v = Simplify[b[m]]]; v];
A[m_] := A[m] = Module[{v}, If[ m == 0, v = 1]; If[m == 1, v = 0]; If[2 <= m , v = -(m - 1)!*Sum[A[k + 1]*B[m - 1 - k]/(k!*(m - 1 - k)!), {k , 0 , m - 2}] + B[m]]; v];
Table[Numerator[A[n]], {n, 0, 15}] (* Jean-François Alcover, Aug 17 2020, translated from Maple *)

A330860 Denominator of the rational number A(n) that appears in the formula for the n-th cumulant k(n) = (-1)^n2^n(A(n) - (n - 1)!*zeta(n)) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, with A(0) = 1 and A(1) = 0.

Original entry on oeis.org

1, 1, 4, 8, 144, 3456, 172800, 10368000, 3810240000, 177811200000, 9957427200000, 75278149632000000, 1912817782149120000000, 53023308921173606400000000, 17742659631203112173568000000000, 426249654980023566857797632000000000, 9600207854287580784554747166720000000000

Offset: 0

Petros Hadjicostas, Apr 28 2020

Hennequin conjectured his cumulant formula in his 1989 paper and proved it in his 1991 thesis.
First he calculates the numbers (B(n): n >= 0), with B(0) = 1 and B(0) = 0, given for p >= 0 by the recurrence
Sum_{r=0..p} Stirling1(p+2, r+1)*B(p-r)/(p-r)! + Sum_{r=0..p} F(r)*F(p-r) = 0, where F(r) = Sum_{i=0..r} Stirling1(r+1,i+1)*G(r-i) and G(k) = Sum_{a=0..k} (-1)^a*B(k-a)/(a!*(k-a)!*2^a).
Then A(n) = L_n(B(1),...,B(n)), where L_n(x_1,...,x_n) are the logarithmic polynomials of Bell.
Hoffman and Kuba (2019, 2020) gave an alternative proof of Hennequin's cumulant formula and gave an alternative calculation for the constants (-2)^n*A(n), which they denote by a_n. See also Finch (2020).
The Maple program given in A330852 is based on Tan and Hadjicostas (1993), where the numbers (A(n): n >= 0) are also tabulated.
For a list of references about the theory of the limiting distribution of the number of comparisons in quicksort, which we do not discuss here, see the ones for sequence A330852.

			The first few fractions A(n) are
1, 0, 7/4, 19/8, 937/144, 85981/3456, 21096517/172800, 7527245453/10368000, 19281922400989/3810240000, 7183745930973701/177811200000, ...
The first few fractions (-2)^n*A(n) (= a_n in Hoffman and Kuba and in Finch) are
1, 0, 7, -19, 937/9, -85981/108, 21096517/2700, -7527245453/81000, 19281922400989/14883750, -7183745930973701/347287500, ...

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991), p. 83.

Petros Hadjicostas, Table of n, a(n) for n = 0..30
S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the first eight moments for the number of comparisons in quicksort from which Hennequin's first eight asymptotic cumulants can be derived.]
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives the constants a_s = (-2)^s*A(s) for s >= 2.]
P. Hennequin, Combinatorial analysis of the quicksort algorithm, Informatique théoretique et applications, 23(3) (1989), 317-333.
M. E. Hoffman and M. Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020; see Section 5.2. [They study the constants a_s = (-2)^s*A(s) for s >= 2.]
Kok Hooi Tan and Petros Hadjicostas, Density and generating functions of a limiting distribution in quicksort, Technical Report #568, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA, 1993; see p. 10.

Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A330852 (numerators).

# The function A is defined in A330852.
# Produces the sequence of denominators of the A(n)'s.
seq(denom(A(n)), n = 0 .. 40);

A329001 Numerator of the rational constant term c(n) of the n-th moment mu(n) of the limiting distribution of the number of comparisons in quicksort (for n >= 0).

Original entry on oeis.org

1, 0, 7, -19, 2260, -229621, 74250517, -30532750703, 90558126238639, -37973078754146051, 21284764359226368337, -1770024989560214080011109, 539780360793818428471498394131, -194520883210026428577888559667954807, 911287963487139630688627952818633149408727

Offset: 0

Petros Hadjicostas, Apr 30 2020

Let X_n be the number of comparisons needed to sort a list of n distinct numbers by quicksort. Then E(X_n) = A115107(n)/A096620(n) = -4*n + + 2*(1+n)*HarmonicNumber(n) or E(X_n) = A093418(n)/A096620(n) = -3*n + + 2*(1+n)*HarmonicNumber(n) depending on the assumptions.
Régnier (1989) and Rösler (1991) proved that (X_n - E(X_n))/n converges a.s. (and in other modes of convergence) to a non-degenerate r.v. X. Rösler proved the existence of all moments for X and Tan and Hadjicostas (1995) proved that it has a density w.r.t. to the Lebesgue measure. Fill and Janson (2000, 2002) proved many other properties of the distribution of X.
Hennequin (1989, 1991) calculated moments and cumulants of X. He proved that mu(n) = E(X^n) is a linear combination of a finite number of zeta values at positive integer arguments with rational coefficients plus a rational constant c(n). It is the numerators of this constant term c(n) of mu(n) that we tabulate here, while the denominators are in A330876.
Hennequin (1991) proved that the n-th cumulant of X for n >= 2 is k(n) = (-2)^n*(A(n) - (n-1)!*zeta(n)), where A(n) = A330852(n)/A330860(n) with A(0) = 1 and A(1) = 0. Also, (-2)^n*A(n) = A330875(n)/A330876(n); see Hoffman and Kuba (2019-2020) and Finch (2020).
Actually, Hoffman and Kuba (2019-2020, Proposition 17) express the constants c(n) in terms of "tiered binomial coefficients". Finch (2020) uses their results to write a Mathematica program with which he calculates at least c(2) - c(8). The values c(2) - c(8) have also been calculated indirectly by Ekhad and Zeilberger (2019).
Because moments and cumulants are connected via the relationship mu(n) = Sum_{s=0..n-1} binomial(n-1,s)*k(s+1)*mu(n-1-s) (e.g., see Tan and Hadjicostas (1993)), we easily deduce that c(n) = Sum_{s=0..n-1} binomial(n-1,s)*(-2)^(s+1)*A(s+1)*c(n-1-s) for n >= 1 because c(1) = A(1) = mu(1) = k(1) = 0 and k(n) = (-2)^n*A(n) - (-2)^n*(n-1)!*zeta(n) for n >= 2 and c(n) is the constant term of mu(n).

			The first few c(n) are {1, 0, 7, -19, 2260/9, -229621/108, 74250517/2700, -30532750703/81000, 90558126238639/14883750, -37973078754146051/347287500, ...}.

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991).

Petros Hadjicostas, Table of n, a(n) for n = 0..25
S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the first eight moments for the number of comparisons in quicksort from which the first eight asymptotic moments mu(n) can be derived.]
James A. Fill and Svante Janson, Smoothness and decay properties of the limiting Quicksort density function, In: D. Gardy and A. Mokkadem (eds), Mathematics and Computer Science, Trends in Mathematics, Birkhäuser, Basel, 2000, pp. 53-64.
James A. Fill and Svante Janson, Quicksort asymptotics, Journal of Algorithms, 44(1) (2002), 4-28.
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives a Mathematica program to generate the constants c(n).]
P. Hennequin, Combinatorial analysis of the quicksort algorithm, Informatique théoretique et applications, 23(3) (1989), 317-333. [He derives some of the asymptotic moments mu(n) whose constant terms are the c(n)'s.]
M. E. Hoffman and M. Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020; see Section 5.2. [They give a formula for the constants c(n).]
Mireille Régnier, A limiting distribution for quicksort, Informatique théorique et applications, 23(3) (1989), 335-343.
Uwe Rösler, A limit theorem for quicksort, Informatique théorique et applications, 25(1) (1991), 85-100. [In Section 5, he gives a recursion for the moments mu(n), which potentially can be used to find the constants c(n), but he does not do any explicit calculations.]
Kok Hooi Tan and Petros Hadjicostas, Density and generating functions of a limiting distribution in quicksort, Technical Report #568, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA, 1993; see pp. 9-11.
Kok Hooi Tan and Petros Hadjicostas, Some properties of a limiting distribution in Quicksort, Statistics and Probability Letters, 25(1) (1995), 87-94.
Vytas Zacharovas, On the exponential decay of the characteristic function of the quicksort distribution, arXiv:1605.04018 [math.CO], 2016.

Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A330852, A330860, A330875, A330876 (denominators).

# The function A is defined in A330852.
# Produces the constants c(n):
c := proc(p) local v, a: option remember:
if p = 0 then v := 1; end if: if p = 1 then v := 0: end if:
if 2 <= p then v := (p - 1)!*add((-2)^(a + 1)*A(a + 1)*c(p - 1 - a)/(a!*(p - 1 - a)!), a = 0 .. p - 1): end if:
v: end proc:
# Produces the numerators of c(n):
seq(numer(c(n)), n=0..15);

(* The function A is defined in A330852. *)
c[p_] := c[p] = Module[{v, a}, If[p == 0, v = 1;]; If[p == 1, v = 0]; If[2 <= p, v = (p - 1)!*Sum[(-2)^(a + 1)*A[a + 1]*c[p - 1 - a]/(a!*(p - 1 - a)!), {a, 0, p - 1}]]; v];
Table[Numerator[c[n]], {n, 0, 15}] (* Jean-François Alcover, Mar 28 2021, after Maple code *)

Recurrence for the fractions: c(n) = Sum_{s=0..n-1} binomial(n-1,s)*(-2)^(s+1)*A(s+1)*c(n-1-s) for n >= 1, with c(0) = 1 = A(0) and c(1) = 0 = A(1), where A(n) = A330852(n)/A330860(n) and (-2)^n*A(n) = A330875(n)/A330876(n).

A330876 Denominator of the fraction fr(n) that appears in the n-th cumulant k(n) = fr(n) - (-2)^n(n-1)!zeta(n) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, starting with fr(0) = 1 and fr(1) = 0.

Original entry on oeis.org

1, 1, 1, 1, 9, 108, 2700, 81000, 14883750, 347287500, 9724050000, 36756909000000, 466996528845000000, 6472571889791700000000, 1082926002881049327000000000, 13008107146607164515924000000000

Offset: 0

Petros Hadjicostas, Apr 29 2020

Hennequin conjectured his cumulant formula in his 1989 paper and proved it in his 1991 thesis.
First he calculates the numbers (B(n): n >= 0), with B(0) = 1 and B(0) = 0, given for p >= 0 by the recurrence
Sum_{r=0..p} Stirling1(p+2, r+1)*B(p-r)/(p-r)! + Sum_{r=0..p} F(r)*F(p-r) = 0, where F(r) = Sum_{i=0..r} Stirling1(r+1,i+1)*G(r-i) and G(k) = Sum_{a=0..k} (-1)^a*B(k-a)/(a!*(k-a)!*2^a).
Then fr(n) = (-2)^n*L_n(B(1),...,B(n)), where L_n(x_1,...,x_n) are the logarithmic polynomials of Bell.
Hoffman and Kuba (2019, 2020) gave an alternative proof of Hennequin's cumulant formula and gave an alternative calculation for the constants fr(n), which they denote by a_n. See also Finch (2020).
Tan and Hadjicostas (1993) used a Maple program (an update of which can be found in A330852) to tabulate the numbers (fr(n)/(-2)^n: n >= 0).
Sequence A330852 contains additional references that give the theory of the limiting distribution of the number of comparisons in quicksort (and for that reason we omit any discussion of that topic).

			The first few fractions fr(n) are: 1, 0, 7, -19, 937/9, -85981/108, 21096517/2700, -7527245453/81000, 19281922400989/14883750, -7183745930973701/347287500, ...

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991), p. 83.

Petros Hadjicostas, Table of n, a(n) for n = 0..30
S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the first eight moments for the number of comparisons in quicksort from which Hennequin's first eight asymptotic cumulants can be derived.]
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives the constants a_s = fr(n) for s >= 2.]
P. Hennequin, Combinatorial analysis of the quicksort algorithm, Informatique théoretique et applications, 23(3) (1989), 317-333. [He made the first conjectures about fr(n).]
M. E. Hoffman and M. Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020; see Section 5.2. [They study the constants a_s = fr(n) for s >= 2.]
Kok Hooi Tan and Petros Hadjicostas, Density and generating functions of a limiting distribution in quicksort, Technical Report #568, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA, 1993; see p. 10 for the constants A(n) = fr(n)/(-2)^n.

Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A329001, A330852, A330860, A330875 (numerators), A330907, A330895.

a(n) = denominator((-2)^n*A330852(n)/A330860(n)).

A330875 Numerator of the fraction fr(n) that appears in the n-th cumulant k(p) = fr(n) - (-2)^n(n-1)!zeta(n) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, starting with fr(0) = 1 and fr(1) = 0.

Original entry on oeis.org

1, 0, 7, -19, 937, -85981, 21096517, -7527245453, 19281922400989, -7183745930973701, 3616944955616896387, -273304346447259998403709, 76372354431694636659849988531, -25401366514997931592208126670898607, 110490677504100075544188675746430710672527

Offset: 0

Petros Hadjicostas, Apr 29 2020

Hennequin conjectured his cumulant formula in his 1989 paper and proved it in his 1991 thesis.
First he calculates the numbers (B(n): n >= 0), with B(0) = 1 and B(0) = 0, given for p >= 0 by the recurrence
Sum_{r=0..p} Stirling1(p+2, r+1)*B(p-r)/(p-r)! + Sum_{r=0..p} F(r)*F(p-r) = 0, where F(r) = Sum_{i=0..r} Stirling1(r+1,i+1)*G(r-i) and G(k) = Sum_{a=0..k} (-1)^a*B(k-a)/(a!*(k-a)!*2^a).
Then fr(n) = (-2)^n*L_n(B(1),...,B(n)), where L_n(x_1,...,x_n) are the logarithmic polynomials of Bell.
Hoffman and Kuba (2019, 2020) gave an alternative proof of Hennequin's cumulant formula and gave an alternative calculation for the constants fr(n), which they denote by a_n. See also Finch (2020).
Tan and Hadjicostas (1993) used a Maple program (an update of which can be found in A330852) to tabulate the numbers (fr(n)/(-2)^n: n >= 0).
Sequence A330852 contains additional references that give the theory of the limiting distribution of the number of comparisons in quicksort (and for that reason we omit any discussion of that topic).

			The first few fractions fr(n) are 1, 0, 7, -19, 937/9, -85981/108, 21096517/2700, -7527245453/81000, 19281922400989/14883750, -7183745930973701/347287500, ...

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991), p. 83.

Petros Hadjicostas, Table of n, a(n) for n = 0..30
S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the first eight moments for the number of comparisons in quicksort from which Hennequin's first eight asymptotic cumulants can be derived.]
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives the constants a_s = fr(n) for s >= 2.]
P. Hennequin, Combinatorial analysis of the quicksort algorithm, Informatique théoretique et applications, 23(3) (1989), 317-333. [He made the first conjectures about fr(n).]
M. E. Hoffman and M. Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020; see Section 5.2. [They study the constants a_s = fr(n) for s >= 2.]
Kok Hooi Tan and Petros Hadjicostas, Density and generating functions of a limiting distribution in quicksort, Technical Report #568, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA, 1993; see p. 10 for the constants A(n) = fr(n)/(-2)^n.

Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A329001, A330852, A330860, A330876 (denominators), A330895, A330907.

a(n) = numerator((-2)^n*A330852(n)/A330860(n)).

A330895 Numerator of the variance of the random number of comparisons in quicksort applied to lists of length n.

Original entry on oeis.org

0, 0, 0, 2, 29, 46, 3049, 60574, 160599, 182789, 4913659, 1072364, 975570703, 1039388677, 5491155875, 92211937094, 6954047816459, 7225392149719, 2699717387790739, 2785123121790325, 573031978700759, 84009502802237, 45510401082365873, 46518885869845328

Offset: 0

Petros Hadjicostas, May 01 2020

			The variances are: 0, 0, 0, 2/9, 29/36, 46/25, 3049/900, 60574/11025, 160599/19600, 182789/15876, 4913659/317520, 1072364/53361, ... = A330895/A330907.

D. E. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching, Addison-Wesley, 1973; see answer to Ex. 8(a) of Section 6.2.2.

S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019; see Theorem 2.
V. Iliopoulos, The quicksort algorithm and related topics, arXiv:1503.02504 [cs.DS], 2015.
V. Iliopoulos and D. Penman, Variance of the number of comparisons of randomized quicksort, arXiv:1006.4063 [math.PR], 2010.

Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A329001, A330852, A330860, A330875, A330876, A330907 (denominators).

lista(nn) = {my(va = vector(nn)); for(n=1, nn, va[n] = numerator(n*(7*n+13) - 2*(n+1)*sum(k=1, n, 1/k) - 4*(n+1)^2*sum(k=1, n, 1/k^2))); concat(0,va);}

a(n) = numerator(fr(n)), where fr(n) = n*(7*n + 13) - 2*(n + 1)*Sum_{k=1..n} 1/k - 4*(n + 1)^2*Sum_{k=1..n} 1/k^2.

A330907 Denominator of the variance of the random number of comparisons in quicksort applied to lists of length n.

Original entry on oeis.org

1, 1, 1, 9, 36, 25, 900, 11025, 19600, 15876, 317520, 53361, 38419920, 33127380, 144288144, 2029052025, 129859329600, 115831315600, 37529346254400, 33870234994596, 6144260316480, 799769421360, 387088399938240, 355503061748835, 40953952713465792, 37864231428870000, 316002721554520000, 2056839142975402500, 1612561888092715560000

Offset: 0

Petros Hadjicostas, May 01 2020

			The variances are: 0, 0, 0, 2/9, 29/36, 46/25, 3049/900, 60574/11025, 160599/19600, 182789/15876, 4913659/317520, 1072364/53361, ... = A330895/A330907.

D. E. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching, Addison-Wesley, 1973; see answer to Ex. 8(a) of Section 6.2.2.

S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019; see Theorem 2.
V. Iliopoulos, The quicksort algorithm and related topics, arXiv:1503.02504 [cs.DS], 2015.
V. Iliopoulos and D. Penman, Variance of the number of comparisons of randomized quicksort, arXiv:1006.4063 [math.PR], 2010.

Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A329001, A330852, A330860, A330875, A330876, A330895 (numerators).

a := n -> denom(2*(n+1)*(harmonic(n,1) + 2*(n+1)*harmonic(n,2))):
seq(a(n), n=0..28); # Peter Luschny, May 02 2020

lista(nn) = {my(va = vector(nn)); for(n=1, nn, va[n] = denominator(n*(7*n+13) - 2*(n+1)*sum(k=1, n, 1/k) - 4*(n+1)^2*sum(k=1, n, 1/k^2))); concat(1, va); }

a(n) = denominator(fr(n)), where fr(n) = n*(7*n + 13) - 2*(n+1)*Sum_{k=1..n} (1/k) - 4*(n+1)^2*Sum_{k=1..n} (1/k^2).

a(n) = denominator(2*(n+1)*(H(n,1) + 2*(n+1)*H(n,2))), where H(n,s) are the generalized harmonic numbers. - Peter Luschny, May 02 2020

A078903 a(n) = n^2 - Sum_{u=1..n} Sum_{v=1..u} valuation(2*v, 2).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 5, 6, 7, 9, 10, 12, 14, 17, 17, 18, 19, 21, 22, 24, 26, 29, 30, 32, 34, 37, 39, 42, 45, 49, 49, 50, 51, 53, 54, 56, 58, 61, 62, 64, 66, 69, 71, 74, 77, 81, 82, 84, 86, 89, 91, 94, 97, 101, 103, 106, 109, 113, 116, 120, 124, 129, 129, 130, 131, 133, 134

Offset: 1

Benoit Cloitre, Dec 12 2002

nonn

This is a fractal generator sequence. Let Fr(m,n) = m*n - a(n); then the graph of Fr(m,n) for 1 <= n <= 4^(m+1) - 3 presents fractal aspects.

			Fr(1, n) for 1 <= n <= 4^2-3 = 13 gives 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 1.
Fr(2, n) for 1 <= n <= 4^3-3 = 63 gives 2, 4, 5, 7, 8, 9, 9, 11, 12, 13, 13, 14, 14, 14, 13, 15, 16, 17, 17, 18, 18, 18, 17, 18, 18, 18, 17, 17, 16, 15, 13, 15, 16, 17, 17, 18, 18, 18, 17, 18, 18, 18, 17, 17, 16, 15, 13, 14, 14, 14, 13, 13, 12, 11, 9, 9, 8, 7, 5, 4, 2.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, preprint, 2016.
Hsien-Kuei Hwang, Svante Janson, Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms 13:4 (2017), #47.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).

Cf. A078904, A073504.

Equals (1/2) * A076178(n).

[n^2-(&+[ &+[Valuation(2*v,2):v in [1..u]]:u in [1..n]]):n in [1..70]]; //  Marius A. Burtea, Oct 24 2019

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)-1+add(i, i=Bits[Split](n)))
    end:
seq(a(n), n=1..68);  # Alois P. Heinz, Feb 03 2024

Accumulate@Table[DigitCount[n, 2, 1] - 1, {n, 68}] (* Ivan Neretin, Sep 07 2017 *)

a(n)=n^2-sum(u=1,n,sum(v=1,u,valuation(2*v,2)))

def A078903(n): return (n+1)*n.bit_count()-n+(sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1)  # Chai Wah Wu, Nov 12 2024

a(n) = n^2 - Sum_{k=1..n} A005187(k);
a(n) = n^2 - Sum_{u=1..n} Sum_{v=1..u} A001511(v);
a(n+1) - a(n) = A048881(n).
G.f.: 1/(1-x)^2 * ((x(1+x)/(1-x) - Sum_{k>=0} x^2^k/(1-x^2^k))). - Ralf Stephan, Apr 12 2002
a(0) = 0, a(2*n) = a(n) + a(n-1) + n - 1, a(2*n+1) = 2*a(n) + n. Also, a(n) = A000788(n) - n. - Ralf Stephan, Oct 05 2003

A076178 a(n) = 2*n^2 - A077071(n).

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 6, 10, 10, 12, 14, 18, 20, 24, 28, 34, 34, 36, 38, 42, 44, 48, 52, 58, 60, 64, 68, 74, 78, 84, 90, 98, 98, 100, 102, 106, 108, 112, 116, 122, 124, 128, 132, 138, 142, 148, 154, 162, 164, 168, 172, 178, 182, 188, 194, 202, 206, 212, 218, 226, 232

Offset: 0

Benoit Cloitre, Nov 01 2002

nonn

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, preprint, 2016.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms 13:4 (2017), #47.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).

Equals 2 * A078903(n).

Cf. A001105, A067699, A077070, A077071.

a(n)=2*n^2-sum(k=0,n,-valuation(polcoeff(pollegendre(2*n),2*k),2))

def A076178(n): return ((n+1)*n.bit_count()-n<<1)+sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # Chai Wah Wu, Nov 12 2024

a(n) = A001105(n) - A077071(n). - Omar E. Pol, Nov 13 2024

cp's OEIS Frontend

A063090 a(n)/(n*n!) is the average number of comparisons needed to find a node in a binary search tree containing n nodes inserted in a random order.

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A330852 Numerator of the rational number A(n) that appears in the formula for the n-th cumulant k(n) = (-1)^n*2^n*(A(n) - (n - 1)!*zeta(n)) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, with A(0) = 1 and A(1) = 0.

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A330860 Denominator of the rational number A(n) that appears in the formula for the n-th cumulant k(n) = (-1)^n*2^n*(A(n) - (n - 1)!*zeta(n)) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, with A(0) = 1 and A(1) = 0.

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A329001 Numerator of the rational constant term c(n) of the n-th moment mu(n) of the limiting distribution of the number of comparisons in quicksort (for n >= 0).

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A330876 Denominator of the fraction fr(n) that appears in the n-th cumulant k(n) = fr(n) - (-2)^n*(n-1)!*zeta(n) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, starting with fr(0) = 1 and fr(1) = 0.

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A330875 Numerator of the fraction fr(n) that appears in the n-th cumulant k(p) = fr(n) - (-2)^n*(n-1)!*zeta(n) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, starting with fr(0) = 1 and fr(1) = 0.

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A330895 Numerator of the variance of the random number of comparisons in quicksort applied to lists of length n.

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A330852 Numerator of the rational number A(n) that appears in the formula for the n-th cumulant k(n) = (-1)^n2^n(A(n) - (n - 1)!*zeta(n)) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, with A(0) = 1 and A(1) = 0.

A330860 Denominator of the rational number A(n) that appears in the formula for the n-th cumulant k(n) = (-1)^n2^n(A(n) - (n - 1)!*zeta(n)) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, with A(0) = 1 and A(1) = 0.

A330876 Denominator of the fraction fr(n) that appears in the n-th cumulant k(n) = fr(n) - (-2)^n(n-1)!zeta(n) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, starting with fr(0) = 1 and fr(1) = 0.

A330875 Numerator of the fraction fr(n) that appears in the n-th cumulant k(p) = fr(n) - (-2)^n(n-1)!zeta(n) of the limiting distribution of the number of comparisons in quicksort, for n >= 2, starting with fr(0) = 1 and fr(1) = 0.