A091725 -id:A091725 - cp's OEIS frontend

A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).

1, 35, 840, 17640, 352800, 6985440, 139708800, 2854051200, 59935075200, 1298593296000, 29088489830400, 674324082432000, 16183777978368000, 402104637462528000, 10339833534750720000, 275039572024369152000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005461, A005990, A062193.

Cf. A001113, A001620, A091725, A099285.

List([0..15],n->Factorial(n+4)*Binomial(n+6,6)/Factorial(4)); # Muniru A Asiru, Jul 01 2018

[Factorial(n+4)*Binomial(n+6, 6)/Factorial(4): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+4)!*Binomial[n+6,6]/4!, {n, 0, 20}] (* G. C. Greubel, May 12 2018 *)

{ f=6; for (n=0, 100, f*=n + 4; write("b062194.txt", n, " ", f*binomial(n + 6, 6)/24) ) } \\ Harry J. Smith, Aug 02 2009

[binomial(n,6)*factorial (n-2)/factorial (4) for n in range(6, 22)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: (1 + 24*x + 90*x^2 + 80*x^3 + 15*x^4)/(1-x)^11.
a(n) = A062139(n+4, 4).
a(n) = (n+4)!*binomial(n+6, 6)/4!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..n} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-4) = (-1)^n*f(n,4,-7), (n >= 4). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 336*(gamma - Ei(1)) - 96*e + 3524/5, where gamma = A001620, Ei(1) = A091725, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 3264*(gamma - Ei(-1)) - 1920/e - 9464/5, where Ei(-1) = -A099285. (End)

A111598 Lah numbers: a(n) = n!*binomial(n-1,7)/8!.

Original entry on oeis.org

1, 72, 3240, 118800, 3920400, 122316480, 3710266560, 111307996800, 3339239904000, 100919250432000, 3088129063219200, 96012739965542400, 3040403432242176000, 98228418580131840000, 3241537813144350720000

Offset: 8

Wolfdieter Lang, Aug 23 2005

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

G. C. Greubel, Table of n, a(n) for n = 8..440

Column 8 of unsigned A008297 and A111596.
Column 7 of A111597.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n-8)*Binomial(n,8)*Binomial(n-1,7): n in [8..35]]; // G. C. Greubel, May 10 2021

Table[(n-8)!*Binomial[n-1,7]*Binomial[n,8], {n,8,35}] (* G. C. Greubel, May 10 2021 *)

[factorial(n-8)*binomial(n,8)*binomial(n-1,7) for n in (8..35)] # G. C. Greubel, May 10 2021

E.g.f.: ((x/(1-x))^8)/8!.
a(n) = (n!/8!)*binomial(n-1, 8-1).
If we define f(n,i,x) = Sum_{k=i..n}(Sum_{j=i..k} (binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) ) ) then a(n) = (-1)^n*f(n,8,-8), (n>=8). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=8} 1/a(n) = 61096*(gamma - Ei(1)) + 54544*e - 338732/5, where gamma = A001620, Ei(1) = A091725 and e = A001113.
Sum_{n>=8} (-1)^n/a(n) = 2107448*(gamma - Ei(-1)) - 1257760/e - 6080436/5, where Ei(-1) = -A099285. (End)

A136659 Unsigned third column (k=2) of triangle A136656 divided by 4.

Original entry on oeis.org

1, 9, 75, 660, 6300, 65520, 740880, 9072000, 119750400, 1696464000, 25686460800, 414096883200, 7083236160000, 128152088064000, 2445351068160000, 49084865077248000, 1033983353475072000, 22808456326656000000, 525810946517176320000, 12645008187498086400000

Offset: 0

Wolfdieter Lang, Feb 22 2008

Also unsigned second column of triangle A136657 divided by 2.

Charalambos A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, Table 8.3, p. 311, with s=-2, k=2 column/4.

Cf. A001710 (1/2 of unsigned k=1 column of A136657). A136660 (k=3 column divided by 8), A136656.

Cf. A001113, A001620, A091725, A099285.

a[n_] := (n + 8)*(n + 1)*(n + 3)!/48; Array[a, 20, 0] (* Amiram Eldar, Aug 31 2025 *)

a(n) = |A136656(n+2,2)|/4, n>=0.
E.g.f.: (2+6*x-3*x^2)/(2*(1-x)^6) (derived from the one given for the column k=2 under A136656).
a(n) = (n+4)!/2 * sum((k+1)!/(k+4)!,k=1..n), with offset 1. - Gary Detlefs, Jul 27 2010
a(n) = (1/48) * (n+8)*(n+1)*(n+3)!. - Gary Detlefs, Aug 03 2010
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=0} 1/a(n) = 44836/245 - 480*e/7 - 24*gamma/7 + 24*ExpIntegralEi(1)/7, where e = A001113, gamma = A001620, and ExpIntegralEi(1) = A091725.
Sum_{n>=0} (-1)^n/a(n) = 39724/245 - 3120/(7*e) + 24*gamma/7 - 24*ExpIntegralEi(-1)/7, where ExpIntegralEi(-1) = -A099285. (End)

A216119 Number of stretching pairs in all permutations in S_n.

Original entry on oeis.org

0, 0, 0, 2, 30, 360, 4200, 50400, 635040, 8467200, 119750400, 1796256000, 28540512000, 479480601600, 8499883392000, 158664489984000, 3112264995840000, 64023737057280000, 1378644471300096000, 31019500604252160000, 728045925946859520000, 17796678189812121600000

Offset: 1

Emeric Deutsch, Feb 26 2013

nonn

A stretching pair of a permutation p in S_n is a pair (i,j) (1 <= i < j <= n) satisfying p(i) < i < j < p(j). For example, for the permutation 31254 in S_5 the pair (2,4) is stretching because p(2) = 1 < 2 < 4 < p(4) = 5.

			a(4) = 2 because 2143 has 1 stretching (namely (2,3)), 3142 has 1 stretching pair (namely (2,3)), and the other 22 permutations in S_4 have no stretching pairs.

E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted)

Vincenzo Librandi, Table of n, a(n) for n = 1..450
E. Clark and R. Ehrenborg, Explicit expressions for the extremal excedance statistic, European J. Combinatorics, 31 (2010), 270-279.
J. Cooper, E. Lundberg, and B. Nagle, Generalized pattern frequency in large permutations, Electron. J. Combin., 20, (2013), Article P28.

Cf. A005461, A216118, A216120.

Cf. A001113, A001620, A091725, A099285.

Concatenation([0],List([2..22],n->Factorial(n)*(n-2)*(n-3)/24)); # Muniru A Asiru, Nov 29 2018

[Factorial(n)*(n-2)*(n-3) div 24: n in [1..30]]; // Vincenzo Librandi, Nov 29 2018

0, seq((1/24)*factorial(n)*(n-2)*(n-3), n = 2 .. 22);

Join[{0}, Table[n! (n - 2) (n - 3) / 24, {n, 2, 30}]] (* Vincenzo Librandi, Nov 29 2018 *)

a(n) = n!*(n-2)*(n-3)/24.
a(n) = 2*A005461(n-3).
a(n) = Sum_{k>=1} A216118(k).
a(n) = Sum_{k>=1} k*A216120(n,k).
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=4} 1/a(n) = 8*(gamma - Ei(1)) + 8*e - 32/3, where gamma = A001620, Ei(1) = A091725, and e = A001113.
Sum_{n>=4} (-1)^n/a(n) = 16*(gamma - Ei(-1)) - 8/e - 28/3, where Ei(-1) = -A099285. (End)
D-finite with recurrence a(n) +(-n-10)*a(n-1) +4*(2*n+3)*a(n-2) +12*(-n+2)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

A283743 Decimal expansion of Ei(1)/e, where Ei is the exponential integral function.

Original entry on oeis.org

6, 9, 7, 1, 7, 4, 8, 8, 3, 2, 3, 5, 0, 6, 6, 0, 6, 8, 7, 6, 5, 4, 7, 8, 6, 8, 1, 9, 1, 9, 5, 5, 1, 5, 9, 5, 3, 1, 7, 1, 7, 5, 4, 3, 0, 9, 5, 4, 3, 6, 9, 5, 1, 7, 3, 2, 0, 0, 5, 4, 8, 0, 7, 7, 8, 9, 4, 5, 4, 1, 1, 5, 1, 9, 5, 1, 4, 4, 2, 6, 9, 6, 2, 9, 1, 0, 0, 5, 3, 0, 3, 0, 3, 3, 3, 9, 1, 1, 4, 0, 0, 6

Offset: 0

Jean-François Alcover, Mar 15 2017

Can be considered the value of the divergent series -0! - 1! - 2! - ... ; see Lagarias reference Section 2.5. - Harry Richman, Jun 14 2020.

			0.6971748832350660687654786819195515953171754309543695173200548...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, equation 44:5:10 at page 426.

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.
Eric Weisstein's World of Mathematics, Exponential Integral.
Eric Weisstein's World of Mathematics, Subfactorial.

Cf. A000166 (subfactorials), A061382 (Pi/e, the imaginary part of subfactorial(-1)), A091725 (Ei(1)), A073003 (-exp(1)*Ei(-1)).

RealDigits[ExpIntegralEi[1]/E, 10, 102][[1]]

real(-eint1(-1)/exp(1)) \\ Michel Marcus, Jun 15 2020

Equals Re(subfactorial(-1)) = Re(Gamma(0,-1)/e).

Equals Sum_{k=1..oo} (-1)^k*psi(k)/Gamma(k), where psi denotes the digamma function (see Spanier and Oldham). - Stefano Spezia, Jan 04 2025

A355987 a(n) = n! * Sum_{k=1..n} 1/floor(n/k)!.

Original entry on oeis.org

1, 3, 13, 61, 421, 2641, 23521, 203281, 2071441, 22407841, 286403041, 3453468481, 51122111041, 759194916481, 12216117513601, 203300293996801, 3811792426041601, 69634723878720001, 1444704854104512001, 29725332567567436801, 658231789483184716801

Offset: 1

Seiichi Manyama, Jul 22 2022

nonn

Cf. A007489, A081125, A345683, A355886, A355988, A355991.

a[n_] := n! * Sum[1/Floor[n/k]!, {k, 1, n}]; Array[a, 21] (* Amiram Eldar, Jul 22 2022 *)

PARI
```
a(n) = n!*sum(k=1, n, 1/(n\k)!);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1,N, (1-x^k)*(exp(x^k)-1))/(1-x)))

E.g.f.: (1/(1-x)) * Sum_{k>0} (1 - x^k) * (exp(x^k) - 1).
a(n) ~ c * n! * n, where c = 0.59962032... - Vaclav Kotesovec, Aug 03 2022
Conjecture: c = Sum_{k>=1} 1/((k+1)!*k) = 2 - exp(1) - A001620 + A091725. - Vaclav Kotesovec, Sep 24 2023

A062195 Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

Original entry on oeis.org

1, 48, 1512, 40320, 997920, 23950080, 570810240, 13699445760, 333923990400, 8310997094400, 211930425907200, 5548723878297600, 149353151057510400, 4135933413900288000, 117874102296158208000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005990, A005461, A062139, A062193, A062194.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n+5)*Binomial(n+7, 7)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+5)!*Binomial[n+7, 7]/5!, {n, 0, 20}] (* G. C. Greubel, May 12 2018 *)

{ f=24; for (n=0, 100, f*=n + 5; write("b062195.txt", n, " ", f*binomial(n + 7, 7)/120) ) } \\ Harry J. Smith, Aug 02 2009

E.g.f.: N(2;5, x)/(1-x)^13 with N(2;5, x) := Sum_{k=0..5} A062196(5, k)*x^k = 1+35*x+210*x^2+350*x^3+175*x^4+21*x^5.
a(n) = A062139(n+5, 5).
a(n) = (n+5)!*binomial(n+7, 7)/5!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-5) = (-1)^(n-1)*f(n,5,-8), (n>=5). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 1295*(Ei(1) - gamma) + 2170*e - 22813/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 36575*(gamma - Ei(-1)) - 21700/e - 63455/3, where Ei(-1) = -A099285. (End)

A225052 E.g.f. satisfies: A(x) = exp( Integral 1/(1 - x*A(x)) dx ).

Original entry on oeis.org

1, 1, 2, 8, 50, 426, 4606, 60418, 932282, 16547562, 332152614, 7439791314, 183964790514, 4977606096570, 146287199495310, 4640510332052370, 158035939351814250, 5750979655319685834, 222710142933114209526, 9144799526131421284434, 396863889188887568805282

Offset: 0

Paul D. Hanna, Apr 26 2013

nonn

Compare to: W(x) = exp( Integral W(x)/(1 - x*W(x)) dx ), which is satisfied by: W(x) = LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Compare to: C(x) = exp( Integral C(x)^2/(1 - x*C(x)^2) dx ), which is satisfied by: C(x) = (1-sqrt(1-4*x))/(2*x) (Catalan numbers, A000108).

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 50*x^4/4! + 426*x^5/5! +...
where
(1) 1/(1 - x*A(x)) = 1 + x + 4*x^2/2! + 24*x^3/3! + 200*x^4/4! + 2130*x^5/5! + 27636*x^6/6! +...+ n*a(n)*x^n/n! +...
(2) log(A(x)) = x + x^2/2! + 4*x^3/3! + 24*x^4/4! + 200*x^5/5! + 2130*x^6/6! + 27636*x^7/7! +...+ n*a(n)*x^(n+1)/(n+1)! +...
(3) A'(x)/A(x) = 1/(1+x*A(x)) + 2!*x*A(x)/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^2*A(x)^2/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) +... = 1/(1-x*A(x)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..390
Eric Weisstein, MathWorld: Exponential Integral

Cf. A091725.

a = ConstantArray[0,20]; a[[1]]=1; Do[a[[n+1]] = a[[n]] + n!*(a[[n]]/(n-1)! + Sum[a[[i]]*a[[n-i]]/i!/(n-i-1)!,{i,1,n-1}]),{n,1,19}]; Flatten[{1,a}] (* Vaclav Kotesovec, Feb 19 2014 *)
FindRoot[ExpIntegralEi[1/r] - ExpIntegralEi[1] == r*E^(1/r),{r,1/2},WorkingPrecision->50] (* program for numerical value of the radius of convergence r, Vaclav Kotesovec, Feb 19 2014 *)

{a(n)=local(A=1+x);for(i=1,n,A=exp(intformal(1/(1-x*A +x*O(x^n)))));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!  satisfies:
(1) 1/(1 - x*A(x)) = 1 + Sum_{n>=1} n*a(n)*x^n/n!.
(2) log(A(x)) = x + Sum_{n>=1} n*a(n)*x^(n+1)/(n+1)!.
(3) log(A(x)) = Integral Sum_{n>=1} n!*(x*A(x))^(n-1) * Product_{k=1..n} 1/(1 + k*x*A(x)) dx. - Paul D. Hanna, Jun 07 2014
E.g.f. derivative: A'(x) = A(x) / (1-x*A(x)). - Vaclav Kotesovec, Feb 19 2014
a(n) ~ n^(n-1) / (exp(n) * r^(n+1/2)), where r = 0.4271853687986028467... is the root of the equation Ei(1/r) - Ei(1) = r*exp(1/r), where Ei is the Exponential integral. - Vaclav Kotesovec, Feb 19 2014

A282822 a(n) = (n - 4)*n! for n>=0.

Original entry on oeis.org

-4, -3, -4, -6, 0, 120, 1440, 15120, 161280, 1814400, 21772800, 279417600, 3832012800, 56043187200, 871782912000, 14384418048000, 251073478656000, 4623936565248000, 89633231880192000, 1824676506132480000, 38926432130826240000, 868546016919060480000

Offset: 0

Bruno Berselli, Feb 22 2017

Cf. A034865.
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

Table[(n - 4) n!, {n, 0, 30}] (* or *)
RecurrenceTable[{a[0] == -4, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}]

E.g.f.: -(4 - 5*x)/(1 - x)^2.
a(n) = n*a(n-1) + n!, with n>0, a(0)=-4.
a(n) = 2*A034865(n) for n>3.
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=5} 1/a(n) = 313/288 - 5*e/12 - gamma/24 + Ei(1)/24 = 313/288 - (5/12)*A001113 - (1/24)*A001620 + A091725/24.
Sum_{n>=5} (-1)^(n+1)/a(n) = -25/288 + 1/(6*e) + gamma/24 - Ei(-1)/24 = -25/288 - (1/6)*A068985 + (1/24)*A001620 + (1/24)*A099285. (End)

A348573 Decimal expansion of exp(-1) * (Ei(1) - gamma).

Original entry on oeis.org

4, 8, 4, 8, 2, 9, 1, 0, 6, 9, 9, 5, 6, 8, 7, 6, 4, 6, 3, 1, 0, 4, 0, 1, 4, 1, 4, 2, 2, 1, 7, 3, 0, 5, 7, 4, 7, 2, 4, 4, 6, 9, 9, 5, 2, 8, 2, 3, 9, 7, 3, 2, 1, 4, 5, 6, 2, 6, 6, 5, 7, 3, 6, 6, 0, 3, 9, 7, 4, 5, 0, 3, 2, 5, 5, 8, 5, 4, 6, 8, 2, 0, 9, 1, 0, 9, 7, 0, 2, 7, 1, 4, 5, 6, 1, 3, 1, 1, 9, 3, 3, 5, 4, 1, 5, 8, 0, 7, 6, 5

Offset: 0

Ilya Gutkovskiy, Oct 23 2021

			0.48482910699568764631040141422173057472446995282397321...

Cf. A001008, A001113, A001620, A002805, A091725, A229837, A283743, A347952.

RealDigits[Exp[-1] (ExpIntegralEi[1] - EulerGamma), 10, 110] [[1]]

(-real(eint1(-1))-Euler)/exp(1) \\ Michel Marcus, Oct 24 2021

Equals Sum_{k>=1} (-1)^(k+1) * H(k) / k!, where H(k) is the k-th harmonic number.

Equals -Integral_{x=0..1} exp(-x)*log(1-x) dx. - Amiram Eldar, Oct 23 2021

cp's OEIS Frontend

A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).

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A111598 Lah numbers: a(n) = n!*binomial(n-1,7)/8!.

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A136659 Unsigned third column (k=2) of triangle A136656 divided by 4.

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A216119 Number of stretching pairs in all permutations in S_n.

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A283743 Decimal expansion of Ei(1)/e, where Ei is the exponential integral function.

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A355987 a(n) = n! * Sum_{k=1..n} 1/floor(n/k)!.

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A062195 Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

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