A111596 -id:A111596 - cp's OEIS frontend

A001755 Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.

1, 20, 300, 4200, 58800, 846720, 12700800, 199584000, 3293136000, 57081024000, 1038874636800, 19833061248000, 396661224960000, 8299373322240000, 181400588328960000, 4135933413900288000, 98228418580131840000, 2426819753156198400000, 62288373664342425600000

Offset: 4

N. J. A. Sloane

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 4..100

Column 4 of A008297.
Column m=4 of unsigned triangle A111596.
Cf. A053495.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n-1)*Binomial(n, 4)/6: n in [4..30]]; // G. C. Greubel, May 10 2021

Maple
```
A001755 := n-> n!*binomial(n-1,3)/4!;
```

Table[n!Binomial[n-1, 3]/4!, {n, 4, 25}] (* T. D. Noe, Aug 10 2012 *)

[binomial(n,4)*factorial (n-1)/6 for n in range(4, 21)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: ((x/(1-x))^4)/4!.
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,4,-4), (n>=4). - Milan Janjic, Mar 01 2009
D-finite with recurrence (-n+4)*a(n) +n*(n-1)*a(n-1)=0. - R. J. Mathar, Jan 06 2021
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 12*(Ei(1) - gamma + 2*e) - 80, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=4} (-1)^n/a(n) = 156*(gamma - Ei(-1)) - 96/e - 88, where Ei(-1) = -A099285. (End)

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001

A001777 Lah numbers: a(n) = n! * binomial(n-1, 4)/5!.

Original entry on oeis.org

1, 30, 630, 11760, 211680, 3810240, 69854400, 1317254400, 25686460800, 519437318400, 10908183686400, 237996734976000, 5394592659456000, 126980411830272000, 3101950060425216000, 78582734864105472000, 2062796790182768640000, 56059536297908183040000

Offset: 5

N. J. A. Sloane

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 5..100

Column 5 of A008297. Cf. A053495.
Column m=5 of unsigned triangle A111596.
Cf. A001113, A001620, A091725, A099285.

Maple
```
A001777 := n-> n!*binomial(n-1,4)/5!;
```

Table[n! Binomial[n - 1, 4]/5!, {n, 5, 20}] (* T. D. Noe, Aug 10 2012 *)

[binomial(n,5)*factorial (n-1)/factorial (4) for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: ((x/(1-x))^5)/5!.
If we define f(n,i,x) = sum(sum(binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n+1)=(-1)^n*f(n,4,-6), (n>=4). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=5} 1/a(n) = 20*(Ei(1) - gamma) - 200*e + 1555/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=5} (-1)^(n+1)/a(n) = 1460*(gamma - Ei(-1)) - 880/e - 2515/3, where Ei(-1) = -A099285. (End)

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

A187543 Binomial convolutions of the central Lah numbers (A187535).

Original entry on oeis.org

1, 4, 80, 2832, 144576, 9660480, 798468480, 78670609920, 9002061573120, 1173384611804160, 171641216823552000, 27843893955582566400, 4961007038613633638400, 963075987422089673932800, 202333751987206944654950400

Offset: 0

Emanuele Munarini, Mar 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187544, A187545, A187546, A187547, A187548.

a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(binomial(n,k)*a(k)*a(n-k), k=0..n),n=0..12);

a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n, k]a[k]a[n - k], {k, 0, n}], {n, 0, 20}]
CoefficientList[Series[(1/2 + EllipticK[16*x]/Pi)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 06 2019 *)

a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(binomial(n,k)*a(k)*a(n-k),k,0,n),n,0,12);

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..n), where L(n) is a central Lah number.
E.g.f.: (1/2 + 1/Pi*K(16x))^2, where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
Recurrence: (n-1)*n^2*(4*n^2-15*n+13)*a(n) = 4*(n-1)*(48*n^5-292*n^4+672*n^3-747*n^2+399*n-76)*a(n-1) - 32*(96*n^7-1000*n^6+4408*n^5-10628*n^4+15034*n^3-12312*n^2+5265*n-854)*a(n-2) + 1024*(2*n-5)^2*(4*n^2-7*n+2)*(n-2)^4*a(n-3). - Vaclav Kotesovec, Aug 10 2013
a(n) ~ n! * log(n) * 2^(4*n-1) / (Pi^2 * n) * (1 + (gamma + Pi + 4*log(2)) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 06 2019

A187544 Stirling transform (of the second kind) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 38, 1310, 66254, 4428782, 368444078, 36691056110, 4256199137774, 563672814445742, 83921091641375918, 13875375391723852910, 2522552600160248918894, 500141581330626431059502, 107400097037199576065830958

Offset: 0

Emanuele Munarini, Mar 11 2011

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187545, A187546, A187547, A187548.

a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(combinat[stirling2](n,k)*a(k), k=0..n),n=0..12);

a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[StirlingS2[n, k]a[k], {k, 0, n}], {n, 0, 20}]
CoefficientList[Series[1/2 + EllipticK[16*(E^x - 1)]/Pi, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 06 2019 *)

a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(stirling2(n,k)*a(k),k,0,n),n,0,12);

a(n) = sum(S(n,k)*L(k),k=0..n), where S(n,k) are the Stirling numbers of the second kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(16(exp(x)-1)) where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (log(17/16))^n). - Vaclav Kotesovec, Oct 06 2019

A187545 Stirling transform (of the first kind) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 38, 1312, 66408, 4442088, 369791064, 36848702784, 4277191653888, 566809715422464, 84441103242634176, 13970100487593468480, 2541362625439551554880, 504185908064687887996800, 108336183242510523080868480

Offset: 0

Emanuele Munarini, Mar 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187546, A187547, A187548.

lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(abs(combinat[stirling1](n,k))*lahc(k), k=0..n), n=0..20);

lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Abs[StirlingS1[n, k]]*lahc[k], {k, 0, n}], {n, 0, 20}]

lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(abs(stirling1(n,k))*lahc(k),k,0,n),n,0,12);

a(n) = sum(s(n,k)*L(k), k=0..n), where s(n,k) are the (signless) Stirling numbers of the first kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(-16*log(1-x)), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (1 - exp(-1/16))^n). - Vaclav Kotesovec, Apr 10 2018

A187546 Stirling transform (of the first kind, with signs) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 34, 1096, 51984, 3262488, 254943384, 23853046656, 2600024557248, 323588157732096, 45276442446814656, 7035574740347812800, 1202158966644148296000, 224022356544364922931840, 45215509996613004825121920

Offset: 0

Emanuele Munarini, Mar 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187545, A187547, A187548.

lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(combinat[stirling1](n,k)*lahc(k), k=0..n), n=0..20);

lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[StirlingS1[n, k]*lahc[k], {k, 0, n}], {n, 0, 20}]

lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(stirling1(n,k)*lahc(k),k,0,n),n,0,12);

a(n) = sum((-1)^(n-k)*s(n,k)*L(k), k=0..n), where s(n,k) are the (signless) Stirling numbers of the first kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(16*log(1+x)), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (exp(1/16) - 1)^n). - Vaclav Kotesovec, Apr 10 2018

A187547 L(n)H(n+1), product of the central Lah number L(n) and the harmonic number H(n).

Original entry on oeis.org

1, 3, 66, 2500, 134260, 9335088, 796938912, 80671795776, 9446603680800, 1256254443100800, 187033518310129920, 30821040496874234880, 5569495264653352381440, 1095113648992295923200000, 232773183612995427763200000, 53186532693832607435089920000

Offset: 0

Emanuele Munarini, Mar 11 2011

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187545, A187546, A187548.

a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(a(n)*sum(1/k,k=1..n+1),n=0..12);

a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[a[n]HarmonicNumber[n + 1], {n, 0, 20}]

a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(a(n)*sum(1/k,k,1,n+1),n,0,12);

Recurrence:

(n+3)(n+2)(n+1)a(n+2)-4(2n+3)^2(2n+5)(n+1)a(n+1)+16(2n+3)^2(2n+1)^2(n+2)a(n)-144delta(n,0)=0.

A235706 (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 6, 18, 9, 1, 0, 24, 36, 12, 1, 0, 0, 60, 60, 15, 1, 0, 0, 0, 120, 90, 18, 1, 0, 0, 0, 0, 210, 126, 21, 1, 0, 0, 0, 0, 0, 336, 168, 24, 1, 0, 0, 0, 0, 0, 0, 504, 216, 27, 1, 0, 0, 0, 0, 0, 0, 0, 720, 270, 30, 1

Offset: 0

Tom Copeland, Apr 20 2014

The associated Laguerre polynomials n!*Lag(n,3-n,-x) are related to the rook polynomials of a rectangular 3 X n chessboard by R(3,n,x) = n!*x^n*Lag(n,3-n,-1/x), which are also the matching polynomials, or generating function of the number of k-edge matchings, of the complete bipartite graph K(m,n), or biclique (cf. Wikipedia for details).

The formulas here and below can be naturally extended with 3 replaced by any positive integer m. For m = 1 and 2, see unsigned A132013 and A132014. The formulas there can be extrapolated to apply to this matrix.

			Triangle begins:
  1;
  3,  1;
  6,  6,  1;
  6, 18,  9,  1;
  0, 24, 36, 12,  1;
  0,  0, 60, 60, 15, 1;
  ...

Michel Marcus, Rows n = 0..50 of triangle, flattened

Cf. A007318, A008306 for a generalization, A132013, A132014, A132440, A238363, A238385.
....................................
With 0th row: 1
n-th row: n!*Lag(n,3-n,-x)
....................................
1st: 1!*Lag(1,2,-x)  = A062139(1,k,-x)
2nd: 2!*Lag(2,1,-x)  = A105278(2,k,x)
3rd: 3!*Lag(3,0,-x)  = A021009(3,k,-x)
4th: 4!*Lag(4,-1,-x) = A111596(4,k,-x)
5th: 5!*Lag(5,-2,-x) = cf. x^2*A062139(3,k,x)
6th: 6!*Lag(6,-3,-x) = cf. x^3*A062137(3,k,-x)
....................................
n-th row: x^(n-3)*3!*Lag(3,n-3,-x)
....................................
1st: x^(-2)*3!Lag(3,-2,-x) = cf. x^(-2)*[x^2*A062139(1,k,x)]
2nd: x^(-1)*3!Lag(3,-1,-x) = x^(-1)*A111596(3,k,-x)
3rd: x^0*3!Lag(3,0,-x)     = x^0*A021009(3,k,-x)
4th: x^1*3!Lag(3,1,-x)     = x^1*A105278(3,k,x)
5th: x^2*3!Lag(3,2,-x)     = x^2*A062139(3,k,-x)
6th: x^3*3!Lag(3,3,-x)     = x^3*A062137(3,k,-x)

/* As triangle */ [[Binomial(3, n-k)*Factorial(n)/Factorial(k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 28 2017

Table[Binomial[3, n - k] n! / k!, {n, 0, 9}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Jul 28 2017 *)

T(n,k) = binomial(3,n-k)*n!/k!
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 28 2017

row(n) = Vecrev(n!*pollaguerre(n, 3-n, -x)); \\ Michel Marcus, Feb 06 2021

T(n,k) = binomial(3,n-k)*n!/k! = binomial(n,k)*3!/(3-n+k)!.
E.g.f.: exp(y*x)(1+y)^3, so this is an Appell sequence of polynomials with lowering operator L= D= d/dx and raising operator R = x + 3/(1+D).
E.g.f. of inverse matrix is exp(x*y)/(1+y)^3.
Multiply the n-th diagonal of the Pascal matrix A007318 by d(0)=1, d(1)=3, d(2)=6, d(3)=6, and d(n)=0 for n>3 to obtain T.
Row polynomials: n!*Lag(n,3-n,-x) = x^(n-3)*3!*Lag(3,n-3,-x) =
(3!/(3-n)!)*K(-n,3-n+1,-x) where K is Kummer's confluent hypergeometric function (as a limit of n+c as c tends to zero).
T = (I + A132440)^3 = exp[3*(A238385-I)]. I = identity matrix.
Operationally, n!Lag(n,3-n,-:xD:) = x^(n-3)*:Dx:^n*x^(3-n) = x^(-3)*:xD:^n*x^3 = n!*binomial(xD+3,n) = n!*binomial(3,n)*K(-n,3-n+1,-:xD:) where :AB:^n = A^n*B^n for any two operators.
n-th row polynomial: n!*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*Lag(k,3,-x). - Peter Bala, Jul 25 2021

A351641 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct runs using exactly k different symbols.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 8, 12, 4, 1, 0, 1, 17, 28, 22, 5, 1, 0, 1, 26, 81, 68, 35, 6, 1, 0, 1, 45, 177, 251, 135, 51, 7, 1, 0, 1, 76, 410, 704, 610, 236, 70, 8, 1, 0, 1, 121, 906, 2068, 2086, 1266, 378, 92, 9, 1

Offset: 0

Andrew Howroyd, Feb 15 2022

Permuting the symbols will not change the structure.

Equivalently, T(n,k) is the number of restricted growth strings [s(0), s(1), ..., s(n-1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1, the maximum value is k and all runs are distinct.

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  5,   3,   1;
  0, 1,  8,  12,   4,   1;
  0, 1, 17,  28,  22,   5,  1;
  0, 1, 26,  81,  68,  35,  6, 1;
  0, 1, 45, 177, 251, 135, 51, 7, 1;
  ...
The T(4,1) = 1 word is 1111.
The T(4,2) = 5 words are 1112, 1121, 1122, 1211, 1222.
The T(4,3) = 3 words are 1123, 1223, 1233.
The T(4,4) = 1 word is 1234.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A351642.
Partial row sums include A000007, A000012, A351018, A351644.
Column k=3 is A351643.
Cf. A111596, A351637, A351640.

\\ here LahI is A111596 as row polynomials.
LahI(n, y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p, i, y)*LahI(i, y))}
R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
T(n)={my(q=S(n), v=concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/r!) ))); [Vecrev(p) | p<-v]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

T(n,k) = A351640(n,k)/k!.

A001778 Lah numbers: a(n) = n!*binomial(n-1,5)/6!.

Original entry on oeis.org

1, 42, 1176, 28224, 635040, 13970880, 307359360, 6849722880, 155831195520, 3636061228800, 87265469491200, 2157837063782400, 55024845126451200, 1447576694865100800, 39291367432052736000, 1100158288097476608000, 31767070568814637056000

Offset: 6

N. J. A. Sloane

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 6..100

Column 6 of A008297.
Column m=6 of unsigned triangle A111596.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n-6)*Binomial(n,6)*Binomial(n-1,5): n in [6..30]]; // G. C. Greubel, May 10 2021

A001778 := proc(n)
    n!*binomial(n-1,5)/6! ;
end proc:
seq(A001778(n),n=6..30) ; # R. J. Mathar, Jan 06 2021

With[{c=6!},Table[n!Binomial[n-1,5]/c,{n,6,24}]] (* Harvey P. Dale, May 25 2011 *)

[binomial(n,6)*factorial(n-1)/factorial(5) for n in range(6, 22)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: ((x/(1-x))^6)/6!.
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,6,-6), (n>=6). - Milan Janjic, Mar 01 2009
D-finite with recurrence (-n+6)*a(n) +n*(n-1)*a(n-1)=0. - R. J. Mathar, Jan 06 2021
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=6} 1/a(n) = 570*(gamma - Ei(1)) + 1380*e - 2999, where gamma = A001620, Ei(1) = A091725 and e = A001113.
Sum_{n>=6} (-1)^n/a(n) = 15030*(gamma - Ei(-1)) - 9000/e - 8661, where Ei(-1) = -A099285. (End)

More terms from Christian G. Bower, Dec 18 2001

cp's OEIS Frontend

A001755 Lah numbers: a(n) = n! * binomial(n-1, 3)/4!.

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A001777 Lah numbers: a(n) = n! * binomial(n-1, 4)/5!.

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A187543 Binomial convolutions of the central Lah numbers (A187535).

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Maple

Mathematica

Maxima

Formula

A187544 Stirling transform (of the second kind) of the central Lah numbers (A187535).

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Maple

Mathematica

Maxima

Formula

A187545 Stirling transform (of the first kind) of the central Lah numbers (A187535).

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Maple

Mathematica

Maxima

Formula

A187546 Stirling transform (of the first kind, with signs) of the central Lah numbers (A187535).

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Maple

Mathematica

Maxima

Formula

A187547 L(n)H(n+1), product of the central Lah number L(n) and the harmonic number H(n).

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Maple

Mathematica