A259870 -id:A259870 - cp's OEIS frontend

A260948 Coefficients in asymptotic expansion of sequence A259870.

1, 2, 5, 17, 74, 395, 2526, 19087, 168603, 1723065, 20148031, 266437102, 3938754720, 64391209604, 1152961464743, 22424127879610, 470399253269776, 10579865622308851, 253840801521314095, 6468953273455413674, 174452533187403980841, 4962228907578051232358

Offset: 0

Vaclav Kotesovec, Aug 05 2015

nonn

			A259870(n)/((n-1)!/exp(1)) ~ 1 + 2/n + 5/n^2 + 17/n^3 + 74/n^4 + 395/n^5 + ...

Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A259870, A260578, A260949, A260950.

nmax = 25; b = CoefficientList[Assuming[Element[x, Reals], Series[x/(ExpIntegralEi[1 + 1/x]/Exp[1 + 1/x] - 1)^2, {x, 0, nmax+1}]], x]; Table[Sum[b[[k+1]]*StirlingS2[n, k-1], {k, 1, n+1}], {n, 0, nmax}]

a(k) ~ 2 * exp(-1) * (k-1)! / (log(2))^k.

A259869 a(0) = -1; for n > 0, number of indecomposable derangements, i.e., no fixed points, and not fixing [1..j] for any 1 <= j < n.

Original entry on oeis.org

-1, 0, 1, 2, 8, 40, 244, 1736, 14084, 128176, 1292788, 14313272, 172603124, 2252192608, 31620422980, 475350915656, 7618759828388, 129697180826512, 2337145267316500, 44446207287450968, 889595868295057364, 18693361200724345024, 411475140936880082020

Offset: 0

N. J. A. Sloane, Jul 09 2015

The derangement characterization would yield a(0) = 1, but -1 is the value given in Martin and Kearney's paper. - Eric M. Schmidt, Jul 10 2015

			There are 9 derangements of 1,2,3,4. All of them are indecomposable except for 2,1,4,3. Thus a(4) = 8. - _Eric M. Schmidt_, Jul 10 2015

Eric M. Schmidt, Table of n, a(n) for n = 0..300
Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A000166, A259870, A259871, A259872, A260578.

Clear[a]; a[0]=-1; a[1]=0; a[n_]:=a[n]=(n-1)*a[n-1] + (n-3)*a[n-2] + Sum[a[j]*a[n-j],{j,1,n-1}]; Table[a[n],{n,0,20}] (* Vaclav Kotesovec, Jul 29 2015 *)
nmax = 25; CoefficientList[Assuming[Element[x, Reals], Series[-x*E^(1 + 1/x)/ExpIntegralEi[1 + 1/x], {x, 0, nmax}]], x] (* Vaclav Kotesovec, Aug 05 2015 *)

def a(n) : return -1 if n==0 else 0 if n==1 else (n-1)*a(n-1) + (n-3)*a(n-2) + sum(a(j)*a(n-j) for j in [1..n-1]) # Eric M. Schmidt, Jul 10 2015

Martin and Kearney (2015) give both a recurrence and a g.f.
The recurrence is a(0)=-1, a(1)=0; a(n) = (n-1)*a(n-1) + (n-3)*a(n-2) + Sum_{j=1..n-1} a(j)*a(n-j).
a(n) ~ n!/exp(1) * (1 - 2/n^2 - 6/n^3 - 29/n^4 - 196/n^5 - 1665/n^6 - 16796/n^7 - 194905/n^8 - 2549468/n^9 - 37055681/n^10), for coefficients see A260578. - Vaclav Kotesovec, Jul 28 2015
G.f.: -1 + x^2/(1 - 2*x - 4*x^2/(1 - 4*x - 9*x^2/(1 - 6*x - 16*x^2/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 22 2018

More terms from and definition edited by Eric M. Schmidt, Jul 10 2015

A259871 a(0)=1/2, a(1)=1; a(n) = na(n-1) + (n-2)a(n-2) - 2Sum_{j=1..n-1} a(j)a(n-j) + 2Sum_{j=0..n-1} a(j)a(n-1-j).

Original entry on oeis.org

1, 2, 5, 14, 45, 170, 777, 4350, 29513, 236530, 2179133, 22576206, 258821269, 3245286490, 44115311969, 645664173566, 10117122765905, 168922438409826, 2993228077070645, 56090022818326542, 1108099905463382973, 23015655499699484810, 501356717394207256441

Offset: 1

N. J. A. Sloane, Jul 09 2015

The sequence officially starts with a(0)=1/2, but since the OEIS only uses integers, we show it with offset 1.

Eric M. Schmidt, Table of n, a(n) for n = 1..300
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A259869, A259870, A259872, A260949.

nmax = 25; Rest[CoefficientList[Assuming[Element[x, Reals], Series[-1/(2*ExpIntegralEi[1 + 1/x]/Exp[1 + 1/x] - 1)/2, {x, 0, nmax}]], x]] (* Vaclav Kotesovec, Aug 05 2015 *)

@CachedFunction
def a(n) : return 1 if n==1 else 2 if n==2 else (n+2)*a(n-1) + (n-2)*a(n-2) - 2*sum(a(j)*a(n-j) for j in [1..n-1]) + 2*sum(a(j)*a(n-1-j) for j in [1..n-2])

Martin and Kearney (2015) give a g.f.

a(n) ~ (n-1)! / exp(1) * (1 + 4/n + 16/n^2 + 76/n^3 + 416/n^4 + 2576/n^5 + 17840/n^6 + 137268/n^7 + 1170104/n^8 + 11050940/n^9 + 115885968/n^10), for coefficients see A260949. - Vaclav Kotesovec, Jul 29 2015

More terms from Eric M. Schmidt, Jul 10 2015

The offset 1 is correct. - N. J. A. Sloane, Jun 16 2021

A259872 a(0)=-1, a(1)=1; a(n) = na(n-1) + (n-2)a(n-2) + Sum_{j=1..n-1} a(j)a(n-j) + 2Sum_{j=0..n-1} a(j)*a(n-1-j).

Original entry on oeis.org

-1, 1, -1, 2, -1, 9, 26, 201, 1407, 11714, 107983, 1102433, 12332994, 150103585, 1974901951, 27935229074, 422799610943, 6818164335881, 116717210194218, 2113959805887881, 40388891717569887, 811833598825134258, 17126091132964548335, 378335451153341591041

Offset: 0

N. J. A. Sloane, Jul 09 2015

Eric M. Schmidt, Table of n, a(n) for n = 0..300
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A259869, A259870, A259871, A260950, A052186.

nmax = 25; CoefficientList[Assuming[Element[x, Reals], Series[-1/(ExpIntegralEi[1 + 1/x]/Exp[1 + 1/x] + 1), {x, 0, nmax}]], x] (* Vaclav Kotesovec, Aug 05 2015 *)

@CachedFunction
def a(n) : return -1 if n==0 else 1 if n==1 else n*a(n-1) + (n-2)*a(n-2) + sum(a(j)*a(n-j) for j in [1..n-1]) + 2*sum(a(j)*a(n-1-j) for j in [0..n-1]) # Eric M. Schmidt, Jul 10 2015

Martin and Kearney (2015) give a g.f.

a(n) ~ (n-1)! / exp(1) * (1 - 2/n + 1/n^2 + 1/n^3 - 10/n^4 - 61/n^5 - 382/n^6 - 3489/n^7 - 39001/n^8 - 484075/n^9 - 6619449/n^10), for coefficients see A260950. - Vaclav Kotesovec, Jul 29 2015

Definition corrected by and more terms from Eric M. Schmidt, Jul 10 2015

A302557 Expansion of 1/(2 - Sum_{k>=0} k!*x^k/(1 + x)^(k+1)).

Original entry on oeis.org

1, 0, 1, 2, 10, 48, 288, 1984, 15660, 139312, 1380484, 15080152, 180017780, 2331038048, 32537274756, 486942025288, 7777172706308, 132025174277392, 2373753512469972, 45059504242538328, 900498975768121972, 18898334957168597184, 415537355533831049572, 9552918187197519923176

Offset: 0

Ilya Gutkovskiy, Aug 15 2018

nonn

Invert transform of A000166.

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Subfactorial

Cf. A000166, A051295, A051296, A259869, A259870, A305275.

nmax = 23; CoefficientList[Series[1/(2 - Sum[k! x^k/(1 + x)^(k + 1), {k, 0, nmax}]), {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 - Sum[Round[k!/Exp[1]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Subfactorial[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

G.f.: 1/(1 - Sum_{k>=1} A000166(k)*x^k).
G.f.: 1/(2 - 1/(1 - x^2/(1 - 2*x - 4*x^2/(1 - 4*x - 9*x^2/(1 - 6*x - 16*x^2/(1 - ...)))))), a continued fraction.
a(n) ~ exp(-1) * n! * (1 + 2/n^2 + 6/n^3 + 35/n^4 + 256/n^5 + 2187/n^6 + 21620/n^7 + 243947/n^8 + 3098528/n^9 + 43799819/n^10 + ...), for coefficients see A305275. - Vaclav Kotesovec, Aug 18 2018

cp's OEIS Frontend

A260948 Coefficients in asymptotic expansion of sequence A259870.

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A259869 a(0) = -1; for n > 0, number of indecomposable derangements, i.e., no fixed points, and not fixing [1..j] for any 1 <= j < n.

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A259871 a(0)=1/2, a(1)=1; a(n) = n*a(n-1) + (n-2)*a(n-2) - 2*Sum_{j=1..n-1} a(j)*a(n-j) + 2*Sum_{j=0..n-1} a(j)*a(n-1-j).

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A259872 a(0)=-1, a(1)=1; a(n) = n*a(n-1) + (n-2)*a(n-2) + Sum_{j=1..n-1} a(j)*a(n-j) + 2*Sum_{j=0..n-1} a(j)*a(n-1-j).

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Crossrefs

Programs

Mathematica

Sage

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Extensions

A302557 Expansion of 1/(2 - Sum_{k>=0} k!*x^k/(1 + x)^(k+1)).

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Formula

A259871 a(0)=1/2, a(1)=1; a(n) = na(n-1) + (n-2)a(n-2) - 2Sum_{j=1..n-1} a(j)a(n-j) + 2Sum_{j=0..n-1} a(j)a(n-1-j).

A259872 a(0)=-1, a(1)=1; a(n) = na(n-1) + (n-2)a(n-2) + Sum_{j=1..n-1} a(j)a(n-j) + 2Sum_{j=0..n-1} a(j)*a(n-1-j).