A091725 -id:A091725 - cp's OEIS frontend

A111599 Lah numbers: a(n) = n!*binomial(n-1,8)/9!.

1, 90, 4950, 217800, 8494200, 309188880, 10821610800, 371026656000, 12614906304000, 428906814336000, 14668613050291200, 506733905373696000, 17735686688079360000, 630299019222512640000, 22780807409042242560000

Offset: 9

Wolfdieter Lang, Aug 23 2005

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

Column 9 of unsigned A008297 and A111596.
Column 8: A111598.
Cf. A001113, A001620, A091725, A099285.

part_ZL:=[S,{S=Set(U,card=r),U=Sequence(Z,card>=1)}, labeled]: seq(count(subs(r=9,part_ZL),size=m),m=9..23) ; # Zerinvary Lajos, Mar 09 2007

Table[n!*Binomial[n-1, 8]/9!, {n, 9, 30}] (* Wesley Ivan Hurt, Dec 10 2013 *)

E.g.f.: ((x/(1-x))^9)/9!.
a(n) = (n!/9!)*binomial(n-1, 9-1).
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-1)*f(n,9,-9), n >= 9. - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=9} 1/a(n) = 564552*(Ei(1) - gamma) - 264528*e - 873657/35, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=9} (-1)^(n+1)/a(n) = 28393416*(gamma - Ei(-1)) - 16938720/e - 573537159/35, where Ei(-1) = -A099285. (End)

A111600 Lah numbers: a(n) = n!*binomial(n-1,9)/10!.

Original entry on oeis.org

1, 110, 7260, 377520, 17177160, 721440720, 28857628800, 1121325004800, 42890681433600, 1629845894476800, 61934143990118400, 2364758225077248000, 91043191665474048000, 3543681152517682176000, 139722285442125754368000

Offset: 10

Wolfdieter Lang, Aug 23 2005

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

Column 10 of unsigned A008297 and A111596.
Column 9: A111599.
Cf. A001113, A001620, A091725, A099285.

Table[n! * Binomial[n - 1, 9]/10!, {n, 10, 25}] (* Amiram Eldar, May 02 2022 *)

E.g.f.: ((x/(1-x))^10)/10!.
a(n) = (n!/10!)*binomial(n-1, 10-1).
If we define f(n,i,x) = Sum_{k=1..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i)*x^(k-j) then a(n) = (-1)^n*f(n,10,-10), (n>=10). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=10} 1/a(n) = 5086710*(gamma - Ei(1)) + 50940*e + 91914449/14, where gamma = A001620, Ei(1) = A091725 and e = A001113.
Sum_{n>=10} (-1)^n/a(n) = 413689770*(gamma - Ei(-1)) - 246749400/e - 3342795017/14, where Ei(-1) = -A099285. (End)

A219541 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} log(1 + k*x).

Original entry on oeis.org

1, 1, 3, 20, 242, 4584, 124936, 4638360, 225037200, 13820428368, 1048006461024, 96171381464256, 10503700943629824, 1346451508974957696, 200184649396819872768, 34167655864475762390784, 6635466680845611611326464, 1454780635849943337186155520

Offset: 0

Paul D. Hanna, Nov 22 2012

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 20*x^3/3! + 242*x^4/4! + 4584*x^5/5! + ...
where
A(x) = 1 + log(1+x) + log(1+x)*log(1+2*x) + log(1+x)*log(1+2*x)*log(1+3*x) + log(1+x)*log(1+2*x)*log(1+3*x)*log(1+4*x) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..144

Cf. A091725, A001620.

a:=series(add(mul(log(1+k*x),k=1..n),n=0..100),x=0,18): seq(n!*coeff(a,x,n),n=0..17); # Paolo P. Lava, Mar 27 2019

With[{nmax = 30}, CoefficientList[Series[Sum[Product[Log[1 + j*x], {j, 1, k}], {k,0,3*nmax}], {x,0,nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 04 2018 *)

{a(n)=n!*polcoeff(sum(m=0,n,prod(k=1,m,log(1+k*x+x*O(x^n)))),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ exp(1/2) * d^(n+1) * (n!)^2, where d = 1/(Ei(1)-gamma) = 1/(A091725 - A001620) = 0.75878167350772..., where Ei is the second exponential integral and gamma is the Euler-Mascheroni constant. - Vaclav Kotesovec, Nov 02 2014

A224788 E.g.f. satisfies: A(x) = exp( Integral A(x)/(1 - x*A(x)^2) dx ).

Original entry on oeis.org

1, 1, 3, 18, 168, 2142, 34704, 682740, 15810372, 421339176, 12702393792, 427435993512, 15881634963216, 645804320863680, 28527455317884336, 1360332028008819360, 69645942884911181184, 3810436222004101378656, 221867131720533800409216, 13698420738298341356760768

Offset: 0

Paul D. Hanna, Apr 28 2013

nonn

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).

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!.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 18*x^3/3! + 168*x^4/4! + 2142*x^5/5! +...
where
log(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 99*x^4/4! + 1236*x^5/5! + 19752*x^6/6! +...
A(x)/(1-x*A(x)^2) = 1 + 2*x + 11*x^2/2! + 99*x^3/3! + 1236*x^4/4! + 19752*x^5/5! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Eric Weisstein, MathWorld: Exponential Integral

Cf. A225052, A091725.

a = ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+2]] = n!*Sum[a[[i+1]]*a[[n-i+1]]/i!/(n-i)!,{i,0,n}] + n!*Sum[a[[j+1]]/(j-1)!*Sum[a[[i+1]]*a[[n-j-i+1]]/i!/(n-j-i)!,{i,0,n}],{j,1,n}],{n,1,18}]; a (* Vaclav Kotesovec, Feb 19 2014 *)
FindRoot[ExpIntegralEi[1/Sqrt[r]] - ExpIntegralEi[1] + E == (r+Sqrt[r]) * E^(1/Sqrt[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(A/(1-x*A^2 +x*O(x^n)))));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

E.g.f. derivative: A'(x) = A(x)^2 / (1-x*A(x)^2). - Vaclav Kotesovec, Feb 19 2014

a(n) ~ n^(n-1) / (sqrt(2) * exp(n) * r^(n+1/4)), where r = 0.28396034297... is the root of the equation Ei(1/sqrt(r)) - Ei(1) + exp(1) = (r+sqrt(r)) * exp(1/sqrt(r)), where Ei is the Exponential Integral. - Vaclav Kotesovec, Feb 19 2014

A306770 Decimal expansion of Sum_{k>=0} 1/(k! + (k+1)! + (k+2)!).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Mar 09 2019

			0.40037967700464134050027...

Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A001113 (exp(1)), A001620 (gamma), A054119, A091725 (ExpIntegralEi[1]).

exp(1) - 1 + Euler - real(-eint1(-1)) \\ Michel Marcus, Mar 09 2019

Sum_{k>=0} 1/(k! + (k+1)! + (k+2)!) = exp(1) - 1 + gamma - ExpIntegralEi[1].
From Amiram Eldar, Jun 26 2021: (Start)
Equals Sum_{k>=2} 1/A054119(k).
Equals -Integral{x=0..1} x*log(x)*exp(x) dx. (End)

A369883 Decimal expansion of Integral_{x=0..1} x/(1 - log(x)) dx.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Feb 04 2024

			0.361328616888222584697161657678739938954590641...

Michael Ian Shamos, A catalog of the real numbers (2011), p. 397.
Eric Weisstein's World of Mathematics, Exponential Integral.
Wikipedia, Exponential Integral.

Cf. A073003, A091725, A099285.

RealDigits[-E^2 * ExpIntegralEi[-2], 10, 120][[1]] (* Amiram Eldar, Feb 04 2024 *)

PARI
```
intnum(x=0,1,x/(1-log(x)))
```

Equals Integral_{x=0..1} x/(1 - log(x)) dx.
Equals - e^2*Ei(-2), where Ei(x) is the Exponential Integral function [Shamos].
Equals Integral_{x=0..oo} dx/(e^x*(x + 2)) [Shamos].

A364521 Decimal expansion of the solution to Ei(x) = x.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Aug 15 2023

Fixed point of exponential integral.

			0.5276123472017420...

Eric Weisstein's World of Mathematics, Exponential Integral.
Wikipedia, Exponential integral.

Cf. A091723, A091725, A099285.

Maple
```
Digits:=101: fsolve(Ei(1,x)-x, x);
```

RealDigits[FindRoot[ExpIntegralE[1, x] - x, {x, 0.5}, WorkingPrecision -> 101][[1, 2]], 10, 101][[1]]

default(realprecision, 101); solve(x=0.5,0.6,eint1(x)-x)

solve(x=0.5,0.6,-Euler()-log(x)-suminf(k=1,(-x)^k/(k*k!))-x)

cp's OEIS Frontend

A111599 Lah numbers: a(n) = n!*binomial(n-1,8)/9!.

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A111600 Lah numbers: a(n) = n!*binomial(n-1,9)/10!.

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A219541 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} log(1 + k*x).

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A224788 E.g.f. satisfies: A(x) = exp( Integral A(x)/(1 - x*A(x)^2) dx ).

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A306770 Decimal expansion of Sum_{k>=0} 1/(k! + (k+1)! + (k+2)!).

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A369883 Decimal expansion of Integral_{x=0..1} x/(1 - log(x)) dx.

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A364521 Decimal expansion of the solution to Ei(x) = x.

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