A260578 -id:A260578 - cp's OEIS frontend

A260503 Coefficients in an asymptotic expansion of sequence A003319.

1, -2, -1, -5, -32, -253, -2381, -25912, -319339, -4388949, -66495386, -1100521327, -19751191053, -382062458174, -7924762051957, -175478462117633, -4132047373455024, -103115456926017761, -2718766185148876961, -75529218928863243200, -2205316818199975235447

Offset: 0

Vaclav Kotesovec, Jul 27 2015

sign

			A003319(n) / n! ~ 1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A003319, A259472, A261214, A261239, A261253, A261254.

Cf. A256168, A260491, A260530, A260532, A260578.

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

a(k) ~ -k! / (2 * (log(2))^(k+1)).
For n>0, Sum_{k=1..n} a(k) * Stirling1(n-1, k-1) = A259472(n). - Vaclav Kotesovec, Aug 03 2015
For n>0, a(n) = Sum_{k=1..n} A259472(k) * Stirling2(n-1, k-1). - Vaclav Kotesovec, Aug 03 2015

A256168 Coefficients in asymptotic expansion of sequence A052186.

Original entry on oeis.org

1, -2, 1, -1, -9, -59, -474, -4560, -50364, -625385, -8622658, -130751886, -2163331779, -38793751015, -749691306018, -15535914341831, -343749787006758, -8089725377931547, -201801866906374263, -5319643146604299835, -147774950436327236681

Offset: 0

Vaclav Kotesovec, Mar 17 2015

sign

For k > 2 is a(k) negative.

			A052186(n) / n! ~ 1 - 2/n + 1/n^2 - 1/n^3 - 9/n^4 - 59/n^5 - 474/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..394
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A052186, A260491, A260503, A260530, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x) / (ExpIntegralEi[1/x] + E^(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-1)! / (log(2))^k.

A260491 Coefficients in asymptotic expansion of sequence A077607.

Original entry on oeis.org

1, -4, 0, -8, -76, -752, -8460, -107520, -1522124, -23717424, -402941324, -7407988448, -146479479308, -3099229422352, -69863683041868, -1671667534710720, -42318672085310540, -1130167625049525232, -31758424368739424780, -936840101208573355680

Offset: 0

Vaclav Kotesovec, Jul 27 2015

sign

For k > 2 is a(k) negative.

			A077607(n) / (-n!) ~ 1 - 4/n - 8/n^3 - 76/n^4 - 752/n^5 - 8460/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..126
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A077607, A111529, A256168, A260503, A260530, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[x^4*E^(2/x)/(ExpIntegralEi[1/x] - x*E^(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 * k! / (4 * (log(2))^(k+2)).

A260532 Coefficients in asymptotic expansion of sequence A051295.

Original entry on oeis.org

1, 2, 7, 31, 165, 1025, 7310, 59284, 543702, 5618267, 65200918, 846462826, 12229783811, 195394019337, 3427472046792, 65526442181293, 1355785469986828, 30166624979467869, 717769036033944699, 18174105506247664633, 487655384740384445407, 13816406622559942660420

Offset: 0

Vaclav Kotesovec, Jul 28 2015

nonn

			A051295(n)/(n-1)! ~ 1 + 2/n + 7/n^2 + 31/n^3 + 165/n^4 + 1025/n^5 + 7310/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..134
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A051295, A134378, A256168, A260491, A260503, A260530, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x)*x / (ExpIntegralEi[1/x] - E^(1/x))^2, {x, 0, nmax+1}]], x]; Table[Sum[b[[k+1]] * StirlingS2[n, k-1], {k, 1, n+1}], {n, 0, nmax}] (* Vaclav Kotesovec, Aug 03 2015 *)

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

a(n) = Sum_{k=0..n} A134378(k) * Stirling2(n, k). - Vaclav Kotesovec, Aug 04 2015

A260530 Coefficients in asymptotic expansion of sequence A051296.

Original entry on oeis.org

1, 2, 7, 35, 216, 1575, 13243, 126508, 1359437, 16312915, 217277446, 3194459333, 51557948291, 908431129702, 17376289236947, 358847480175063, 7959468559605624, 188702262366570387, 4760773506835189975, 127312428854513811012, 3596091234340397964321

Offset: 0

Vaclav Kotesovec, Jul 28 2015

nonn

			A051296(n) / n! ~ 1 + 2/n + 7/n^2 + 35/n^3 + 216/n^4 + 1575/n^5 + 13243/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. A051296, A256168, A260491, A260503, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x)*x^2 / (ExpIntegralEi[1/x] - 2*x*E^(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 * (log(2))^(k+1)).

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

A260948 Coefficients in asymptotic expansion of sequence A259870.

Original entry on oeis.org

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.

A260949 Coefficients in asymptotic expansion of sequence A259871.

Original entry on oeis.org

1, 4, 16, 76, 416, 2576, 17840, 137268, 1170104, 11050940, 115885968, 1353366864, 17640817784, 256630492660, 4153220868128, 74315436120300, 1458541231513152, 31131651836906752, 716862465409883040, 17683184383300077828, 464519709712796199816

Offset: 0

Vaclav Kotesovec, Aug 05 2015

nonn

			A259871(n)/((n-1)!/exp(1)) ~ 1 + 4/n + 16/n^2 + 76/n^3 + 416/n^4 + 2576/n^5 + ...

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

Cf. A259871, A260578, A260948, A260950.

nmax = 25; b = CoefficientList[Assuming[Element[x, Reals], Series[x/(2*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) ~ 4 * exp(-1) * (k-1)! / (log(2))^k.

A260950 Coefficients in asymptotic expansion of sequence A259872.

Original entry on oeis.org

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.

A305275 Coefficients in asymptotic expansion of sequence A302557.

Original entry on oeis.org

1, 0, 2, 6, 35, 256, 2187, 21620, 243947, 3098528, 43799819, 682540780, 11630529643, 215190967544, 4296657514283, 92083313483300, 2108244638675035, 51350077108834832, 1325682930813985547, 36157047428501464220, 1038793351537388253211, 31354977545074731373512

Offset: 0

Vaclav Kotesovec, Aug 18 2018

nonn

			A302557(n) / (exp(-1) * n!) ~ 1 + 2/n^2 + 6/n^3 + 35/n^4 + 256/n^5 + 2187/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..75

Cf. A256168, A260491, A260503, A260530, A260532, A260578, A260948, A260949, A260950, A302557.

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

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A260503 Coefficients in an asymptotic expansion of sequence A003319.

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A256168 Coefficients in asymptotic expansion of sequence A052186.

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A260491 Coefficients in asymptotic expansion of sequence A077607.

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A260532 Coefficients in asymptotic expansion of sequence A051295.

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A260530 Coefficients in asymptotic expansion of sequence A051296.

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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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A260948 Coefficients in asymptotic expansion of sequence A259870.

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A260949 Coefficients in asymptotic expansion of sequence A259871.

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