cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-46 of 46 results.

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

Original entry on oeis.org

1, 90, 4950, 217800, 8494200, 309188880, 10821610800, 371026656000, 12614906304000, 428906814336000, 14668613050291200, 506733905373696000, 17735686688079360000, 630299019222512640000, 22780807409042242560000
Offset: 9

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

References

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

Crossrefs

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

Programs

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

Formula

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

Views

Author

Wolfdieter Lang, Aug 23 2005

Keywords

References

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

Crossrefs

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

Programs

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

Formula

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)

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

Views

Author

Claude H. R. Dequatre, Feb 04 2024

Keywords

Examples

			0.361328616888222584697161657678739938954590641...

Links

Crossrefs

Cf. A073003, A091725, A099285.

Programs

  • Mathematica
    RealDigits[-E^2 * ExpIntegralEi[-2], 10, 120][[1]] (* Amiram Eldar, Feb 04 2024 *)
  • PARI
    intnum(x=0,1,x/(1-log(x)))

Formula

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].

A377400 Decimal expansion of e*(gamma - Ei(-1))/2.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, Oct 27 2024

Keywords

Examples

			1.08269110766346817971049317424621528419071038...

Links

Crossrefs

Cf. A000142, A001008, A001113, A001620, A005843, A060746, A099285, A347952, A377401.

Programs

  • Mathematica
    RealDigits[E(EulerGamma-ExpIntegralEi[-1])/2,10,100][[1]]

Formula

Equals Sum_{n>=1} (1/n!)*Sum_{k=1..n} 1/(2*k) (see Shamos).
Equals A347952 / 2.

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

Views

Author

Michal Paulovic, Aug 15 2023

Keywords

Comments

Fixed point of exponential integral.

Examples

			0.5276123472017420...

Links

Crossrefs

Cf. A091723, A091725, A099285.

Programs

  • Maple
    Digits:=101: fsolve(Ei(1,x)-x, x);
  • Mathematica
    RealDigits[FindRoot[ExpIntegralE[1, x] - x, {x, 0.5}, WorkingPrecision -> 101][[1, 2]], 10, 101][[1]]
  • PARI
    default(realprecision, 101); solve(x=0.5,0.6,eint1(x)-x)
  • PARI
    solve(x=0.5,0.6,-Euler()-log(x)-suminf(k=1,(-x)^k/(k*k!))-x)

A371667 Decimal expansion of -Ei(-1) / log(2).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 10 2024

Keywords

Examples

			0.316504114203126786893754620753862815669085943...

Crossrefs

Cf. A007525, A019704, A099285, A232734.

Programs

  • Mathematica
    RealDigits[-ExpIntegralEi[-1]/Log[2], 10, 103][[1]]
  • PARI
    eint1(1)/log(2) \\ Michel Marcus, Apr 11 2024

Formula

Equals Integral_{x=0..oo} exp(-2^x) dx.
Previous Showing 41-46 of 46 results.

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