A111596 -id:A111596 - cp's OEIS frontend

A111597 Lah numbers: a(n) = n!*binomial(n-1,6)/7!.

1, 56, 2016, 60480, 1663200, 43908480, 1141620480, 29682132480, 779155977600, 20777492736000, 565147802419200, 15721384321843200, 448059453172531200, 13097122477350912000, 392913674320527360000, 12101741169072242688000

Offset: 7

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 = 7..440

Column 7 of A008297 and unsigned A111596.
Column 6 of A001778.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n-7)*Binomial(n, 7)*Binomial(n-1, 6): n in [7..30]]; // G. C. Greubel, May 10 2021

k = 7; a[n_] := n!*Binomial[n-1, k-1]/k!; Table[a[n], {n, k, 22}]  (* Jean-François Alcover, Jul 09 2013 *)

[factorial(n-7)*binomial(n, 7)*binomial(n-1, 6) for n in (7..30)] # G. C. Greubel, May 10 2021

E.g.f.: ((x/(1-x))^7)/7!.
a(n) = (n!/7!)*binomial(n-1, 7-1).
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) = (-1)^n*f(n,6,-8), (n>=6). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=7} 1/a(n) = 6342*(Ei(1) - gamma) - 8988*e + 80374/5, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=7} (-1)^(n+1)/a(n) = 170142*(gamma - Ei(-1)) - 101640/e - 490714/5, where Ei(-1) = -A099285. (End)

A180047 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 + w/(...)))).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 6, 1, 0, 24, 6, 0, 120, 36, 1, 0, 720, 240, 12, 0, 5040, 1800, 120, 1, 0, 40320, 15120, 1200, 20, 0, 362880, 141120, 12600, 300, 1, 0, 3628800, 1451520, 141120, 4200, 30, 0, 39916800, 16329600, 1693440, 58800, 630, 1, 0, 479001600

Offset: 0

Wouter Meeussen, Aug 08 2010

Equivalence to the binomial formula needs formal proof. This c.f. converges to A052119 = 0.697774657964.. = BesselI(1,2)/BesselI(0,2) for w = 1.

			Triangle starts:
  0;
  0,   1;
  0,   2;
  0,   6,   1;
  0,  24,   6;
  0, 120,  36,  1;
  0, 720, 240, 12;
The numerator of w/(1+w/(2+w/(3+w/(4+w/5)))) equals 120*w + 36*w^2 + w^3.

Variant: A221913.

Cf. A084950, A180048, A180049, A008297, A111596, A105278, A052119.

Table[CoefficientList[Numerator[Together[Fold[w/(#2+#1) &,Infinity,Reverse @ Table[k,{k,1,n}]]]],w],{n,16}]; (* or equivalently *) Table[(n-m+1)!/m! *Binomial[n-m,m-1], {n,0,16}, {m,0,Floor[n/2+1/2]}]

T(n,m) = (n-m+1)!/m!*binomial(n-m, m-1) for n >= 0, 0 <= m <= (n+1)/2.

A351640 Triangle read by rows: T(n,k) is the number of patterns of length n with all distinct runs and maximum value k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 10, 18, 24, 0, 1, 16, 72, 96, 120, 0, 1, 34, 168, 528, 600, 720, 0, 1, 52, 486, 1632, 4200, 4320, 5040, 0, 1, 90, 1062, 6024, 16200, 36720, 35280, 40320, 0, 1, 152, 2460, 16896, 73200, 169920, 352800, 322560, 362880

Offset: 0

Andrew Howroyd, Feb 15 2022

A pattern is a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670.

			Triangle begins:
  1,
  0, 1;
  0, 1,  2;
  0, 1,  4,   6;
  0, 1, 10,  18,  24;
  0, 1, 16,  72,  96, 120;
  0, 1, 34, 168, 528, 600, 720;
  ...
The T(3,1) = 1 pattern is 111.
The T(3,2) = 4 patterns are 112, 122, 211, 221.
The T(3,3) = 6 patterns are 123, 132, 213, 231, 312, 321.

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

Row sums are A351200.
Main diagonal is A000142.
Cf. A000670, A111596, A350824, A351641.

\\ 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)) ))); [Vecrev(p) | p<-v]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

T(n,k) = k! * A351641(n,k).

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)

A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 10, 1, 0, 100, 140, 28, 1, 0, 1700, 2900, 840, 60, 1, 0, 44200, 85800, 31460, 3300, 110, 1, 0, 1635400, 3476200, 1501500, 203060, 10010, 182, 1, 0, 81770000, 185874000, 90563200, 14700400, 943800, 25480, 280, 1

Offset: 0

Peter Luschny, Mar 07 2016

0

			[1]
[0,        1]
[0,        2,         1]
[0,       10,        10,        1]
[0,      100,       140,       28,        1]
[0,     1700,      2900,      840,       60,      1]
[0,    44200,     85800,    31460,     3300,    110,     1]
[0,  1635400,   3476200,  1501500,   203060,  10010,   182,   1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A269946 (order 3).

Cf. A036969, A105278, A204579, A269938, A269941, A269944, A269945.

T := proc(n,k) option remember;
if n=k then return 1 fi; if k<0 or k>n then return 0 fi;
T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k) end:
seq(seq(T(n,k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A268434_row := n -> PTrans(n, n->`if`(n=1,1, ((n-1)^2+1)/(n*(4*n-2))),
(n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A268434_row(n)), n=0..8);

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^2 + k^2)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

#cached_function
def T(n, k):
    if n==k: return 1
    if k<0 or k>n: return 0
    return T(n-1, k-1)+((n-1)^2+k^2)*T(n-1, k)
for n in range(8): print([T(n, k) for k in (0..n)])
# Alternatively with the function PtransMatrix (cf. A269941):
PtransMatrix(8, lambda n: 1 if n==1 else ((n-1)^2+1)/(n*(4*n-2)), lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k))

T(n,k) = (-1)^k*((2*n)!/(2*k)!)*P[n,k](s(n)) where P is the P-transform and s(n) = ((n-1)^2+1)/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n,k) = Sum_{j=k..n} A269944(n,j)*A269945(j,k).
T(n,1) = Product_{k=1..n} (k-1)^2+1 for n>=1 (cf. A101686).
T(n,n-1) = (n-1)*n*(2*n-1)/3 for n>=1 (cf. A006331).
Row sums: A269938.

A344050 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n, k)|Lah(n, k)|. Inverse binomial convolution of the unsigned Lah numbers A271703.

Original entry on oeis.org

1, 1, -3, 1, 73, -699, 3001, 24697, -783999, 10946233, -80958779, -656003919, 40097528857, -944102982419, 14449693290033, -81180376526759, -4110744092532479, 203618771909117937, -5868277577182238579, 117997016943575159713, -1055340561026036009559, -45279878749358024400299

Offset: 0

Peter Luschny, May 10 2021

sign

Cf. A271703, A111596, A344050.

aList := proc(len) local lah;
lah := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1)*n!/k!):
seq(add((-1)^(n-k)*binomial(n, k)*lah(n, k), k = 0..n), n = 0..len-1) end:
print( aList(22) );

a[n_] := (-1)^(n-1) n n! HypergeometricPFQ[{1 - n, 1 - n}, {2, 2}, -1]; a[0] := 1;
Table[a[n], {n, 0, 20}]

a(n) = (-1)^(n-1)*n*n!*hypergeom([1 - n, 1 - n], [2, 2], -1) for n >= 1.

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

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)

A173227 Partial sums of A000262.

Original entry on oeis.org

1, 2, 5, 18, 91, 592, 4643, 42276, 436629, 5033182, 63974273, 888047414, 13358209647, 216334610860, 3751352135263, 69325155322184, 1359759373992105, 28206375825238458, 616839844140642301, 14181213537729200474, 341879141423814854915, 8623032181189674581256

Offset: 0

Jonathan Vos Post, Feb 13 2010

nonn

Partial sums of the number of "sets of lists": number of partitions of {1,..,n} into any number of lists, where a list means an ordered subset. The subsequence of primes begins: 2, 5, 4643, 616839844140642301.

			a(20) = 1 + 1 + 3 + 13 + 73 + 501 + 4051 + 37633 + 394353 + 4596553 + 58941091 + 824073141 + 12470162233 + 202976401213 + 3535017524403 + 65573803186921 + 1290434218669921 + 26846616451246353 + 588633468315403843 + 13564373693588558173 + 327697927886085654441.

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A001263, A001700, A002868, A008297, A052852, A066668, A111596,.

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[n eq 0 select 1 else 1 + (&+[ Factorial(j)*( l(j,-1) - l(j-1,-1) ): j in [1..n]]): n in [0..25]]; // G. C. Greubel, Mar 09 2021

b:= proc(n) option remember; `if`(n=0, 1, add(
       b(n-j)*j!*binomial(n-1, j-1), j=1..n))
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

With[{m = 25}, CoefficientList[Exp[x/(1-x)] + O[x]^m, x] Range[0, m-1]!// Accumulate] (* Jean-François Alcover, Nov 21 2020 *)
Table[1 +Sum[j!*(LaguerreL[j, -1] -LaguerreL[j-1, -1]), {j,n}], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[1 + sum(factorial(j)*(gen_laguerre(j,0,-1) - gen_laguerre(j-1,0,-1)) for j in (1..n)) for n in (0..30)] # G. C. Greubel, Mar 09 2021

From Vaclav Kotesovec, Oct 25 2016: (Start)
a(n) = 2*n*a(n-1) - (n^2 - n + 1)*a(n-2) + (n-2)*(n-1)*a(n-3).
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/sqrt(2) * (1 - 5/(48*sqrt(n))).
(End)
a(n) = 1 + Sum_{j=1..n} j!*( LaguerreL(j,-1) - LaguerreL(j-1,-1) ). - G. C. Greubel, Mar 09 2021

A216154 Triangle read by rows, T(n,k) n>=0, k>=0, generalization of A000255.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 21, 9, 1, 53, 128, 78, 16, 1, 309, 905, 710, 210, 25, 1, 2119, 7284, 6975, 2680, 465, 36, 1, 16687, 65821, 74319, 35035, 7945, 903, 49, 1, 148329, 660064, 857836, 478464, 133630, 19936, 1596, 64, 1, 1468457, 7275537, 10690812, 6879684, 2279214, 419958, 44268, 2628, 81, 1

Offset: 0

Peter Luschny, Sep 19 2012

			     1,
     1,      1,
     3,      4,      1,
    11,     21,      9,      1,
    53,    128,     78,     16,      1,
   309,    905,    710,    210,     25,      1,
  2119,   7284,   6975,   2680,    465,     36,      1,
16687,  65821,  74319,  35035,   7945,    903,     49,      1,
148329, 660064, 857836, 478464, 133630,  19936,   1596,     64,      1,

A000255 (col. 0), A110450 (diag. n,n-2).

Cf. A111596, A271703.

A216154 := proc(n,k) local L, Z;
L := (n,k) -> `if`(k<0 or k>n,0,(n-k)!*C(n,n-k)*C(n-1,n-k)):
Z := (n,k) -> add(C(-j,-n)*L(j,k), j=0..n);
Z(n+1, k+1) end:
seq(seq(A216154(n,k), k=0..n), n=0..9); # Peter Luschny, Apr 13 2016

T[0, 0] = 1; T[0, ] = 0; T[n, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (2k+1) T[n-1, k] + (k+1) (k+2) T[n-1, k+1]; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

def A216154_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(1+2*k)*M[n-1,k]+(k+1)*(k+2)*M[n-1,k+1]
    return M
A216154_triangle(9)

Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+(1+2*k)*T(n-1,k)+(k+1)*(k+2)*T(n-1,k+1).

Let Z(n, k) = Sum_{j=0..n} C(-j, -n)*L(j, k) where L denotes the unsigned Lah numbers A271703. Then T(n, k) = Z(n+1, k+1). - Peter Luschny, Apr 13 2016

cp's OEIS Frontend

A111597 Lah numbers: a(n) = n!*binomial(n-1,6)/7!.

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A180047 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 + w/(...)))).

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A351640 Triangle read by rows: T(n,k) is the number of patterns of length n with all distinct runs and maximum value k.

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

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A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

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A344050 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*|Lah(n, k)|. Inverse binomial convolution of the unsigned Lah numbers A271703.

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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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A344050 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n, k)|Lah(n, k)|. Inverse binomial convolution of the unsigned Lah numbers A271703.