A259872 -id:A259872 - cp's OEIS frontend

A260950 Coefficients in asymptotic expansion of sequence A259872.

1, -2, 1, 1, -10, -61, -382, -3489, -39001, -484075, -6619449, -99610098, -1638687448, -29255834780, -563343011377, -11639759292186, -256916737692132, -6034068201092777, -150271333127027481, -3955735249215111270, -109757859467421502791

Offset: 0

Vaclav Kotesovec, Aug 05 2015

sign

			A259872(n)/((n-1)!/exp(1)) ~ 1 - 2/n + 1/n^2 + 1/n^3 - 10/n^4 - 61/n^5 - ...

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

Cf. A259872, A260578, A260948, A260949.

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

A259870 a(0)=0, a(1)=1; a(n) = na(n-1) + (n-2)a(n-2) - Sum_{j=1..n-1} a(j)*a(n-j).

Original entry on oeis.org

0, 1, 1, 2, 5, 17, 74, 401, 2609, 19802, 171437, 1664585, 17892938, 210771761, 2698597601, 37301188610, 553473138677, 8773014886289, 147928235322314, 2643635547262049, 49909639472912177, 992516629078846010, 20736210820909594109, 454084963076923193321

Offset: 0

N. J. A. Sloane, Jul 09 2015

Colin Barker, Table of n, a(n) for n = 0..450
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A259869, A259871, A259872, A260948.

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

a=vector(30); a[1]=0; a[2]=1; for(n=2, #a-1, a[n+1] = n*a[n] + (n-2)*a[n-1] - sum(j=1, n-1, a[j+1]*a[n-j+1])); a \\ Colin Barker, Jul 09 2015

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

a(n) ~ (n-1)! / exp(1) * (1 + 2/n + 5/n^2 + 17/n^3 + 74/n^4 + 395/n^5 + 2526/n^6 + 19087/n^7 + 168603/n^8 + 1723065/n^9 + 20148031/n^10), for coefficients see A260948. - Vaclav Kotesovec, Jul 29 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

cp's OEIS Frontend

A260950 Coefficients in asymptotic expansion of sequence A259872.

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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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A259870 a(0)=0, a(1)=1; a(n) = n*a(n-1) + (n-2)*a(n-2) - Sum_{j=1..n-1} a(j)*a(n-j).

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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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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A259870 a(0)=0, a(1)=1; a(n) = na(n-1) + (n-2)a(n-2) - Sum_{j=1..n-1} a(j)*a(n-j).

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).