A271705 -id:A271705 - cp's OEIS frontend

A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.

1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -6, 11, -6, 1, 0, 24, -50, 35, -10, 1, 0, -120, 274, -225, 85, -15, 1, 0, 720, -1764, 1624, -735, 175, -21, 1, 0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1, 0, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1, 0, -362880, 1026576, -1172700, 723680, -269325, 63273, -9450, 870, -45, 1

Offset: 0

N. J. A. Sloane

The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.
Mirror image of the triangle A054654. - Philippe Deléham, Dec 30 2006
Also the triangle gives coefficients T(n,k) of x^k in the expansion of C(x,n) = (a(k)*x^k)/n!. - Mokhtar Mohamed, Dec 04 2012
From Wolfdieter Lang, Nov 14 2018: (Start)
This is the Sheffer triangle of Jabotinsky type (1, log(1 + x)). See the e.g.f. of the triangle below.
This is the inverse Sheffer triangle of the Stirling2 Sheffer triangle A008275.
The a-sequence of this Sheffer triangle (see a W. Lang link in A006232)
is from the e.g.f. A(x) = x/(exp(x) -1) a(n) = Bernoulli(n) = A027641(n)/A027642(n), for n >= 0. The z-sequence vanishes.
The Boas-Buck sequence for the recurrences of columns has o.g.f. B(x) = Sum_{n>=0} b(n)*x^n = 1/((1 + x)*log(1 + x)) - 1/x. b(n) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), b = {-1/2, 5/12, -3/8, 251/720, -95/288, 19087/60480,...}. For the Boas-Buck recurrence of Riordan and Sheffer triangles see the Aug 10 2017 remark in A046521, adapted to the Sheffer case, also for two references. See the recurrence and example below. (End)
Let G(n,m,k) be the number of simple labeled graphs on [n] with m edges and k components. Then T(n,k) = Sum (-1)^m*G(n,m,k). See the Read link below. Equivalently, T(n,k) = Sum mu(0,p) where the sum is over all set partitions p of [n] containing k blocks and mu is the Moebius function in the incidence algebra associated to the set partition lattice on [n]. - Geoffrey Critzer, May 11 2024

			Triangle begins:
  n\k 0     1       2       3      4      5      6    7    8   9 ...
  0   1
  1   0     1
  2   0    -1       1
  3   0     2      -3       1
  4   0    -6      11      -6      1
  5   0    24     -50      35    -10      1
  6   0  -120     274    -225     85    -15      1
  7   0   720   -1764    1624   -735    175    -21    1
  8   0 -5040   13068  -13132   6769  -1960    322  -28    1
  9   0 40320 -109584  118124 -67284  22449  -4536  546  -36   1
  ... - _Wolfdieter Lang_, Aug 22 2012
------------------------------------------------------------------
From _Wolfdieter Lang_, Nov 14 2018: (Start)
Recurrence: s(5,2)= s(4, 1) - 4*s(4, 2) = -6 - 4*11 = -50.
Recurrence from the a- and z-sequences: s(6, 3) = 2*(1*1*(-50) + 3*(-1/2)*35 + 6*(1/6)*(-10) + 10*0*1) = -225.
Boas-Buck recurrence for column k = 3, with b = {-1/2, 5/12, -3/8, ...}:
s(6, 3) = 6!*((-3/8)*1/3! + (5/12)*(-6)/4! + (-1/2)*35/5!) = -225. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245.
J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

T. D. Noe, Rows n=0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.
Fatima Zohra Bensaci, Rachid Boumahdi, and Laala Khaldi, Finite Sums Involving Fibonacci and Lucas Numbers, J. Int. Seq. (2024). See p. 9.
R. M. Dickau, Stirling numbers of the first kind. [Illustrates the unsigned Stirling cycle numbers A132393.]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7.
Gergő Nemes, An Asymptotic Expansion for the Bernoulli Numbers of the Second Kind, J. Int. Seq. 14 (2011), #11.4.8.
A. Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011), #11.8.2 (A-number typo A048894).
NIST Digital Library of Mathematical Functions, Stirling Numbers
Ken Ono, Larry Rolen, and Florian Sprung, Zeta-Polynomials for modular form periods, p. 6, arXiv:1602.00752 [math.NT], 2016.
Ricardo A. Podestá, New identities for binary Krawtchouk polynomials, binomial coefficients and Catalan numbers, arXiv preprint arXiv:1603.09156 [math.CO], 2016.
Ronald Read, An Introduction to Chromatic Polynomials, Journal of Combinatorial Theory, 4(1968)52-71.
Wikipedia, Stirling numbers and exponential generating functions.

See especially A008275 which is the main entry for this triangle. A132393 is an unsigned version, and A008276 is another version.
Cf. A008277, A039814-A039817, A048993, A084938.
A000142(n) = Sum_{k=0..n} |s(n, k)| for n >= 0.
Row sums give A019590(n+1).
Cf. A002209, A027641/A027642, A216294, A271705.

a048994 n k = a048994_tabl !! n !! k
a048994_row n = a048994_tabl !! n
a048994_tabl = map fst $ iterate (\(row, i) ->
(zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 0)
-- Reinhard Zumkeller, Mar 18 2013

A048994:= proc(n,k) combinat[stirling1](n,k) end: # R. J. Mathar, Feb 23 2009
seq(print(seq(coeff(expand(k!*binomial(x,k)),x,i),i=0..k)),k=0..9); # Peter Luschny, Jul 13 2009
A048994_row := proc(n) local k; seq(coeff(expand(pochhammer(x-n+1,n)), x,k), k=0..n) end: # Peter Luschny, Dec 30 2010

Table[StirlingS1[n, m], {n, 0, 9}, {m, 0, n}] (* Peter Luschny, Dec 30 2010 *)

create_list(stirling1(n,k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

a(n,k) = if(k<0 || k>n,0, if(n==0,1,(n-1)*a(n-1,k)+a(n-1,k-1)))

trg(nn)=for (n=0, nn-1, for (k=0, n, print1(stirling(n,k,1), ", ");); print();); \\ Michel Marcus, Jan 19 2015

s(n, k) = A008275(n,k) for n >= 1, k = 1..n; column k = 0 is {1, repeat(0)}.
s(n, k) = s(n-1, k-1) - (n-1)*s(n-1, k), n, k >= 1; s(n, 0) = s(0, k) = 0; s(0, 0) = 1.
The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0) = a(0, k) = 0; a(0, 0) = 1.
Triangle (signed) = [0, -1, -1, -2, -2, -3, -3, -4, -4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; Triangle(unsigned) = [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.
Sum_{k=0..n} (-m)^(n-k)*s(n, k) = A000142(n), A001147(n), A007559(n), A007696(n), ... for m = 1, 2, 3, 4, ... .- Philippe Deléham, Oct 29 2005
A008275*A007318 as infinite lower triangular matrices. - Gerald McGarvey, Aug 20 2009
T(n,k) = n!*[x^k]([t^n]exp(x*log(1+t))). - Peter Luschny, Dec 30 2010, updated Jun 07 2020
From Wolfdieter Lang, Nov 14 2018: (Start)
Recurrence from the Sheffer a-sequence (see a comment above): s(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j, j)*Bernoulli(j)*s(n-1, k-1+j), for n >= 1 and k >= 1, with s(n, 0) = 0 if n >= 1, and s(0,0) = 1.
Boas-Buck type recurrence for column k: s(n, k) = (n!*k/(n - k))*Sum_{j=k..n-1} b(n-1-j)*s(j, k)/j!, for n >= 1 and k = 0..n-1, with input s(n, n) = 1. For sequence b see the Boas-Buck comment above. (End)
T(n,k) = Sum_{j=k..n} (-1)^(n-j)*A271705(n,j)*A216294(j,k). - Mélika Tebni, Feb 23 2023

Offset corrected by R. J. Mathar, Feb 23 2009

Formula corrected by Philippe Deléham, Sep 10 2009

A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 73, 136, 78, 16, 1, 501, 1045, 730, 210, 25, 1, 4051, 9276, 7515, 2720, 465, 36, 1, 37633, 93289, 85071, 36575, 8015, 903, 49, 1, 394353, 1047376, 1053724, 519456, 137270, 20048, 1596, 64, 1, 4596553, 12975561

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (cf. A008297): L'(n,i)=n!/i!*binomial(n-1,i-1) for i >= 1, L'(0,0)=1, L'(n,0)=0 for n>0. T(n,0)=A000262(n); T(n,2)=A052852(n). Row sums A052897.
Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - Paul Barry, Apr 28 2007
From Wolfdieter Lang, Jun 22 2017: (Start)
The inverse matrix T^(-1) is exponential Riordan (aka Sheffer) (e^(-x), x/(1+x)): T^(-1)(n, m) = (-1)^(n-m)*A271705(n, m).
The a- and z-sequences of this Sheffer (aka exponential Riordan) matrix are a = [1,1,repeat(0)] and z(n) = (-1)^(n+1)*A028310(n)/A000027(n-1) with e.g.f. ((1+x)/x)*(1-exp(-x)). For a- and z-sequences see a W. Lang link under A006232 with references. (End)

			The triangle T = A007318*A271703 starts:
n\m       0        1        2       3       4      5     6    7  8 9 ...
0:        1
1:        1        1
2:        3        4        1
3:       13       21        9       1
4:       73      136       78      16       1
5:      501     1045      730     210      25      1
6:     4051     9276     7515    2720     465     36     1
7:    37633    93289    85071   36575    8015    903    49    1
8:   394353  1047376  1053724  519456  137270  20048  1596   64  1
9:  4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1
... reformatted. - _Wolfdieter Lang_, Jun 22 2017
E.g.f. for T(n, 2) = 1/2!*(x/(1-x))^2*e^(x/(x-1)) = 1*x^2/2 + 9*x^3/3! + 78*x^4/4! + 730*x^5/5! + 7515*x^6/6 + ...
From _Wolfdieter Lang_, Jun 22 2017: (Start)
The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...
T recurrence: T(3, 0) = 3*(1*T(2,0) + (1/2)*T(2, 1) + (-2/3)*T(2 ,1)) = 3*(3 + (1/2)*4 - (2/3)) = 13; T(3, 1) = 3*(T(2, 0)/1 + T(2, 1)) = 3*(3 + 4) = 21.
Meixner type recurrence for R(2, x): (D - D^2)*(3 + 4*x + x^2) = 4 + 2*x - 2 = 2*(1 + x), (D = d/dx).
General Sheffer recurrence for R(2, x): (1+x)*(1 + 2*D + D^2)*(1 + x) = (1+x)*(1 + x + 2) = 3 + 4*x + x^2. (End)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A007318, A008297, A052852, A052897, A271703, A271705.

Concatenation([1],Flat(List([1..10],n->List([0..n],m->Sum([0..n],i-> Factorial(n)/Factorial(i)*Binomial(n-1,i-1)*Binomial(i,m)))))); # Muniru A Asiru, Jul 25 2018

A059110:= func< n,k | n eq 0 select 1 else Factorial(n-1)*Binomial(n,k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;
[A059110(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Lprime := proc(n,i)
    if n = 0 and i = 0 then
        1;
    elif k = 0 then
        0 ;
    else
        n!/i!*binomial(n-1,i-1) ;
    end if;
end proc:
A059110 := proc(n,k)
    add(Lprime(n,i)*binomial(i,k),i=0..n) ;
end proc: # R. J. Mathar, Mar 15 2013

(* First program *)
lp[n_, i_] := Binomial[n-1, i-1]*n!/i!; lp[0, 0] = 1; t[n_, m_] := Sum[lp[n, i]*Binomial[i, m], {i, 0, n}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)
(* Second program *)
A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];
Table[A059110[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)
flatten([[A059110(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

E.g.f. for column m: (1/m!)*(x/(1-x))^m*e^(x/(x-1)), m >= 0.
From Wolfdieter Lang, Jun 22 2017: (Start)
E.g.f. for row polynomials in powers of x (e.g.f. of the triangle): exp(z/(1-z))* exp(x*z/(1-z)) (exponential Riordan).
Recurrence: T(n, 0) = Sum_{j=0} z(j)*T(n-1, j), n >= 1, with z(n) = (-1)^(n+1)*A028310(n), T(0, 0) = 1, T(n, m) = 0 n < m, T(n, m) = n*(T(n-1, m-1)/m + T(n-1, m)), n >= m >= 1 (from the z- and a-sequence, see a comment above).
Meixner type recurrence for the (monic) row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: Sum_{k=0..n-1} (-1)^k*D^(k+1)*R(n, x) = n*R(n-1, x), n >=1, R(0, x) = 1, with D = d/dx.
General Sheffer recurrence: R(n, x) = (x+1)*(1+D)^2*R(n-1, x), n >=1, R(0, x) = 1.
(End)
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of this entry; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 18 2018
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (n-1)!*binomial(n, k)*LaguerreL(n-1, 1-k, -1) with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A052897(n). (End)

A308876 Expansion of e.g.f. exp(x)(1 - x)/(1 - 2x).

Original entry on oeis.org

1, 2, 7, 40, 317, 3166, 37987, 531812, 8508985, 153161722, 3063234431, 67391157472, 1617387779317, 42052082262230, 1177458303342427, 35323749100272796, 1130359971208729457, 38432239021096801522, 1383560604759484854775, 52575302980860424481432

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Binomial transform of A002866.

Alois P. Heinz, Table of n, a(n) for n = 0..403

Cf. A002866, A010844, A011782, A067273, A161936, A271705, A294466, A327606.

Row sums of A326659.

a:= n-> n! * add(ceil(2^(n-k-1))/k!, k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2019

nmax = 19; CoefficientList[Series[Exp[x] (1 - x)/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[1 + Sum[Binomial[n,k] 2^(k - 1) k!, {k, 1, n}], {n, 0, 19}]

a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 2^(k-1) * k!.
a(n) = A010844(n) - A067273(n).
a(n) ~ n! * 2^(n-1) * exp(1/2). - Vaclav Kotesovec, Jun 29 2019
a(n) = Sum_{k=0..n} k! * A271705(n,k). - Alois P. Heinz, Sep 12 2019

A326659 T(n,k) = [0=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 15, 18, 6, 1, 64, 132, 96, 24, 1, 325, 980, 1140, 600, 120, 1, 1956, 7830, 12720, 10440, 4320, 720, 1, 13699, 68502, 143850, 162120, 103320, 35280, 5040, 1, 109600, 657608, 1698816, 2447760, 2123520, 1108800, 322560, 40320

Offset: 0

Alois P. Heinz, Sep 12 2019

[] is an Iverson bracket.

			Triangle T(n,k) begins:
  1;
  1,     1;
  1,     4,     2;
  1,    15,    18,      6;
  1,    64,   132,     96,     24;
  1,   325,   980,   1140,    600,    120;
  1,  1956,  7830,  12720,  10440,   4320,   720;
  1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Iverson bracket

Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).
Main diagonal gives A000142.
Row sums give A308876.
Cf. A000522, A073474, A093964, A143409, A196347, A271705, A327606.

T:= proc(n, k) option remember;
      `if`(0=0, 1, 0)
    end:
seq(seq(T(n, k), k=0..n), n=0..10);

T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

E.g.f. of column k: exp(x)*(x/(1-x))^k.
T(n,k) = k! * A271705(n,k).
T(n,k) = n * A073474(n-1,k-1) for n,k >= 1.
T(n,1) = n * A000522(n-1) for n >= 1.
T(n,2) = n * A093964(n-1) for n >= 1.
Sum_{k=1..n} k * T(n,k) = A327606(n).

A134432 Sum of entries in all the arrangements of the set {1,2,...,n} (to n=0 there corresponds the empty set).

Original entry on oeis.org

0, 1, 9, 66, 490, 3915, 34251, 328804, 3452436, 39456405, 488273005, 6510306726, 93097386174, 1421850988831, 23105078568495, 398118276872520, 7251440043035176, 139227648826275369, 2810658160680434001, 59519819873232720010, 1319356007189991960210

Offset: 0

Emeric Deutsch, Nov 16 2007

nonn

Appears to be the binomial transform of A001286 (filled with the appropriate two leading zeros), shifted one index left. - R. J. Mathar, Apr 04 2012

			a(2)=9 because the arrangements of {1,2} are (empty), 1, 2, 12 and 21.

Alois P. Heinz, Table of n, a(n) for n = 0..447

Cf. A000522, A001286, A134431, A271703, A271705, A326659.

[Binomial(n+1,2)*(&+[Factorial(j)*Binomial(n-1, j-1): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 09 2022

Q[0]:=1: for n to 17 do Q[n]:=sort(simplify(Q[n-1]+t^n*x*(diff(x*Q[n-1], x))), t) end do: for n from 0 to 17 do P[n]:=sort(subs(x=1,Q[n])) end do: seq(subs(t =1,diff(P[n],t)),n=0..17);
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, [t!, 0],
      b(n-1, t)+(p-> p+[0, n*p[1]])(b(n-1, t+1)))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 19 2020

(* First program *)
b[n_, s_, t_]:= b[n, s, t] = If[n==0, t! x^s, b[n-1, s, t] + b[n-1, s+n, t+1]];
T[n_]:= T[n]= Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] @ b[n, 0, 0];
a[n_] := Sum[k T[n][[k+1]], {k, 0, n(n+1)/2}];
a /@ Range[0, 20] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz *)
(* Second program *)
a[n_]:= ((n+1)/2)*Sum[j*j!*Binomial[n,j], {j,0,n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 09 2022 *)

[((n+1)/2)*sum( j*factorial(j)*binomial(n, j) for j in (0..n) ) for n in (0..30)] # G. C. Greubel, Jan 09 2022

a(n) = Sum_{k=0..n*(n+1)/2} k*A134431(n,k).
a(n) = (d/dt)P[n](t) evaluated at t=1; here P[n](t)=Q[n](t,1) where the polynomials Q[n](t,x) are defined by Q[0]=1 and Q[n]=Q[n-1] + xt^n (d/dx)xQ[n-1]. (Q[n](t,x) is the bivariate generating polynomial of the arrangements of {1,2,...,n}, where t (x) marks the sum (number) of the entries; for example, Q[2](t,x) = 1 + tx + t^2*x + 2t^3*x^2, corresponding to: empty, 1, 2, 12 and 21, respectively.)
E.g.f.: exp(x)*x*(2 + x - x^2) / (2*(1 - x)^3). - Ilya Gutkovskiy, Jun 02 2020
From G. C. Greubel, Jan 09 2022: (Start)
a(n) = A271705(n+1, 2).
a(n) = ((n+1)/2) * Sum_{j=0..n} j * j! * binomial(n, j).
a(n) = (1/n!)*binomial(n+1, 2) * Sum_{j=0..n} (j!)^2 * A271703(n, j). (End)
D-finite with recurrence (-n+1)*a(n) +(n+1)^2*a(n-1) -n*(n+1)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

More terms from Alois P. Heinz, Dec 22 2017

A271704 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)L(j,k), L the unsigned Lah numbers A271703, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 11, 8, 1, 0, 49, 57, 15, 1, 0, 261, 424, 174, 24, 1, 0, 1631, 3425, 1930, 410, 35, 1, 0, 11743, 30336, 21855, 6320, 825, 48, 1, 0, 95901, 294553, 259161, 95235, 16835, 1491, 63, 1, 0, 876809, 3123632, 3251500, 1452976, 325150, 38864, 2492, 80, 1

Offset: 0

Peter Luschny, Apr 14 2016

			Triangle starts:
  [1]
  [0, 1]
  [0, 3,     1]
  [0, 11,    8,     1]
  [0, 49,    57,    15,    1]
  [0, 261,   424,   174,   24,   1]
  [0, 1631,  3425,  1930,  410,  35,  1]
  [0, 11743, 30336, 21855, 6320, 825, 48, 1]

A001339 (col. 1), A005563 (diag. n,n-1).

Cf. A271703, A271705.

L := (n,k) -> `if`(k<0 or k>n,0,(n-k)!*binomial(n,n-k)*binomial(n-1,n-k)):
T := (n,k) -> add(L(j,k)*binomial(-j,-n)*(-1)^(n-j), j=0..n):
seq(seq(T(n,k), k=0..n), n=0..9);

From Natalia L. Skirrow, Jun 12 2025: (Start)
Definition can also be written Sum_{j=0..n} C(n-1, j-1)*L(j, k), where C(n, -1) = (1 if n = -1 else 0) over integer n.
T(n, k) = C(n-1, k-1)*A143409(n-k, k) = k*(k+1)*A271705(n+1, k+1)/(n*(n+1)) for n > 0.
E.g.f. for sequence b(n, k) = T(n+1, k+1): F/(1-x)^2, where F = exp(x + y*x/(1-x)) is e.g.f. of A271705.
E.g.f. for kth column of b: exp(x)*x^(k-1)/(1-x)^(k+1)/(k-1)!. (These cannot be integrated with x to give e.g.f.s for T(n, k) using standard functions.)
T(n, k) = Sum_{i=0..n-k} (n-1)_i*(i+1)*A271705(n-1-i, k-1), where (n)_i = n!/(n-i)! is the falling factorial. (End)

cp's OEIS Frontend

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A271704 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*L(j,k), L the unsigned Lah numbers A271703, for n>=0 and 0<=k<=n.

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A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

A308876 Expansion of e.g.f. exp(x)(1 - x)/(1 - 2x).

A271704 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)L(j,k), L the unsigned Lah numbers A271703, for n>=0 and 0<=k<=n.