A099285 -id:A099285 - cp's OEIS frontend

A052521 Number of pairs of sequences of cardinality at least 3.

0, 0, 0, 0, 0, 0, 720, 10080, 120960, 1451520, 18144000, 239500800, 3353011200, 49816166400, 784604620800, 13076743680000, 230150688768000, 4268249137152000, 83230858174464000, 1703031405723648000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..445
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 88.

Cf. sequences with formula (n + k)*n! listed in A282466.

Cf. A001113, A001620, A068985, A091725, A099285.

Concatenation([0,0,0,0,0,0], List([6..20], n-> (n-5)*Factorial(n))); # G. C. Greubel, May 13 2019

[n le 5 select 0 else (n-5)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 13 2019

spec := [S,{B=Sequence(Z,3 <= card), S=Prod(B,B)},labeled]: # Pairs spec
seq(combstruct[count](spec, size=n), n=0..20);

Table[If[n<6, 0, (n-5)*n!], {n,0,20}] (* G. C. Greubel, May 13 2019 *)

{a(n) = if(n<6, 0, (n-5)*n!)}; \\ G. C. Greubel, May 13 2019

[0,0,0,0,0,0]+[(n-5)*factorial(n) for n in (6..20)] # G. C. Greubel, May 13 2019

E.g.f.: x^6/(1-x)^2.
(n-5)*a(n+1) + (4 + 3*n - n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 720.
a(n) = (n-5)*n!.
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=6} 1/a(n) = 5477/7200 - 17*e/60 - gamma/120 + Ei(1)/120 = 5477/7200 - (17/60)*A001113 - (1/120)*A001620 + A091725/120.
Sum_{n>=6} (-1)^n/a(n) = 403/7200 - 1/(6*e) + gamma/120 - Ei(-1)/120 = 403/7200 - (1/6)*A068985 + (1/120)*A001620 + (1/120)*A099285. (End)

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

Original entry on oeis.org

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

A257535 Decimal expansion of the imaginary part of -E_1(i), i being the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 28 2015

E_1(z) = Integral_{t>=1}(exp(-t*z)/t) is the exponential integral.

			0.6247132564277136042899683778165717842862467449494411200160175...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Exponential integral.

Cf. A099281, A099282, A099285.

RealDigits[Pi/2 - SinIntegral[1], 10, 105][[1]] (* Amiram Eldar, May 24 2023 *)

PARI
```
a = imag(-eint1(I))
```

Equals imag(E_1(-i)).

Equals (Pi/2) - A099281.

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)

A245780 Decimal expansion of (1-C_2)/e, a constant connected with two-sided generalized Fibonacci sequences, where C_2 is the Euler-Gompertz constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 01 2014

			0.148495506775922047918359994701339218414763837624859626929858...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.2 Euler-Gompertz Constant, p. 426.

Walther Janous, Problem 1552, Crux Mathematicorum, Vol. 16, No. 6 (1990), p. 171; Solution to Problem 1552, by Richard Katz, ibid., Vol. 17, No. 7 (1991), pp. 223-224.
Peter Fishburn, Andrew Odlyzko and Fred Roberts, Two-sided generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 27, No. 4 (1989), pp. 352-361.

Cf. A073003 (C_2), A099285 (C_2 / e).

$RecursionLimit = 10^4; digits = 101; m0 = 100; dm = 100; Clear[g]; g[m_] := g[m] = (Clear[a, b, f]; b[n_] := 2*n; a[n_ /; n >= m] = 0; a[1] = 1; a[2] = -1; a[n_] := -(n-1)^2; f[m] = b[m]; f[n_] := f[n] = b[n] + a[n+1]/f[n+1]; (1 - f[0])/E); g[m0]; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits] != RealDigits[g[m - dm], 10, digits], m = m + dm]; RealDigits[g[m], 10, digits] // First
(* or, as verification: *) RealDigits[1/E + ExpIntegralEi[-1], 10, digits] // First

1/exp(1) - eint1(1,1)[1] \\ Michel Marcus, Aug 06 2020

Equals 1/e + Ei(-1), where Ei is the exponential integral function.
Equals Integral_{x=0..1} exp(-1/x) dx. - Amiram Eldar, Aug 06 2020
Equals Integral_{x=1..+oo} exp(-x)/x^2 dx. - Jianing Song, Oct 03 2021
Equals lim_{n->oo} (Sum_{k=1..n-1} (k/(k+1))^n)/n (Janous, 1990). - Amiram Eldar, Apr 03 2022

A249385 Decimal expansion of gamma - 2*Ei(-1), one of the Tauberian constants, where Ei is the exponential integral function.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 27 2014

			1.01598353369257340796083964100264572910425392275374...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Tauberian Constants, August 30, 2004 [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Exponential Integral

Cf. A001620, A073003, A099285, A248472.

evalf(gamma - 2*Ei(-1), 120); # Vaclav Kotesovec, Oct 27 2014

RealDigits[ EulerGamma - 2*ExpIntegralEi[-1], 10, 103] // First

default(realprecision, 100); Euler + 2*eint1(1) \\ G. C. Greubel, Sep 04 2018

Also equals gamma + 2*G/e, where G is the Euler-Gompertz constant 0.596347...

Equals A001620 + 2*A073003/e. - G. C. Greubel, Sep 04 2018

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)

cp's OEIS Frontend

A052521 Number of pairs of sequences of cardinality at least 3.

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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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A257535 Decimal expansion of the imaginary part of -E_1(i), i being the imaginary unit.

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

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