A259869 -id:A259869 - cp's OEIS frontend

A260578 Coefficients in asymptotic expansion of sequence A259869.

1, 0, -2, -6, -29, -196, -1665, -16796, -194905, -2549468, -37055681, -592013436, -10307671769, -194225544124, -3937581243201, -85460277981116, -1977127315636969, -48573021658496348, -1262954975286604673, -34650561545808167292, -1000438355724912080873

Offset: 0

Vaclav Kotesovec, Jul 29 2015

sign

For k > 1 is a(k) negative.

			A259869(n) / (n!/exp(1)) ~ 1 - 2/n^2 - 6/n^3 - 29/n^4 - 196/n^5 - 1665/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..132
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A259869, A256168, A260491, A260503, A260530, A260532, A260948, A260949, A260950.

nmax = 20; b = CoefficientList[Assuming[Element[x, Reals], Series[x^2*E^(2 + 2/x)/ExpIntegralEi[1 + 1/x]^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

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

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

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

A260578 Coefficients in asymptotic expansion of sequence A259869.

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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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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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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

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

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

Formula

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).

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).