A002720 -id:A002720 - cp's OEIS frontend

A277421 a(n) = n!LaguerreL(n, -7n).

1, 8, 254, 13614, 1025048, 99368620, 11781698256, 1651548277946, 267197019684224, 49000715036948304, 10044513851042988800, 2275926588768085912582, 564838094735322988575744, 152378369304839730672573044, 44397985962782115253758973952

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials

Cf. A277373, A277391, A277392, A277418, A277419, A277420, A277422.

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*7^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Table[n!*LaguerreL[n, -7*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 7^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*7^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 7^k * n^k / k!.

a(n) ~ sqrt(1/2 + 9/(2*sqrt(77))) * ((9 + sqrt(77))/2)^n * exp((-9 + sqrt(77))*n/2) * n^n.

A277422 a(n) = n!LaguerreL(n, -8n).

Original entry on oeis.org

1, 9, 322, 19446, 1649688, 180184120, 24070390992, 3801662863152, 692979602529664, 143184960501077376, 33069665092749868800, 8442378658666161822976, 2360674573114695421197312, 717531421372546588398529536, 235551703250624390582942574592

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

In general, if m > 0 and a(n) = n!*LaguerreL(n, -m*n), then a(n) ~ sqrt(1/2 + (m+2)/(2*sqrt(m*(m+4)))) * (2+m+sqrt(m*(m+4)))^n * exp(n*(sqrt(m*(m+4))-m-2)/2) * n^n / 2^n.

For m > 4, (-1)^n * n! * LaguerreL(n, m*n) ~ sqrt(1/2 + (m-2)/(2*sqrt(m*(m-4)))) * exp((m - 2 - sqrt(m*(m-4)))*n/2) * ((m - 2 + sqrt(m*(m-4)))/2)^n * n^n. - Vaclav Kotesovec, Feb 20 2020

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials

Cf. A277373 (m=1), A277391 (m=2), A277392 (m=3), A277418 (m=4), A277419 (m=5), A277420 (m=6), A277421 (m=7).

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*(8)^k*n^k/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, May 16 2018

Table[n!*LaguerreL[n, -8*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 8^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

for(n=0, 30, print1(n!*sum(k=0,n, binomial(n,k)*(8)^k*n^k/k!), ", ")) \\ G. C. Greubel, May 16 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 8^k * n^k / k!.

a(n) ~ sqrt(2 + 5/sqrt(6)) * (5 + 2*sqrt(6))^n * exp((-5 + 2*sqrt(6))*n) * n^n / 2.

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

Original entry on oeis.org

1, -2, 6, 3, -24, 30, -4, 60, -180, 140, 5, -120, 630, -1120, 630, -6, 210, -1680, 5040, -6300, 2772, 7, -336, 3780, -16800, 34650, -33264, 12012, -8, 504, -7560, 46200, -138600, 216216, -168168, 51480, 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790

Offset: 0

N. J. A. Sloane, Jan 17 2020

Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.

Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - Michael Somos, Mar 20 2020 [Corrected by Petros Hadjicostas, Jul 09 2020]

			Triangle begins:
   1;
  -2,    6;
   3,  -24,    30;
  -4,   60,  -180,     140;
   5, -120,   630,   -1120,     630;
  -6,  210, -1680,    5040,   -6300,     2772;
   7, -336,  3780,  -16800,   34650,   -33264,   12012;
  -8,  504, -7560,   46200, -138600,   216216, -168168,   51480;
   9, -720, 13860, -110880,  450450, -1009008, 1261260, -823680, 218790;
  ...

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Buhl, Book review: J. Ser - Les calculs formels des séries de factorielles, L'Enseignement Mathématique, 32 (1933), p. 275.
L. A. MacColl, Review: J. Ser, Les calculs formels des séries de factorielles, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.
L. M. Milne-Thomson, Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars), The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages.)

Columns 1 is A331433 or equally A007531, column 2 is A331434 or equally A054559; the last three diagonals are A002738, A002736, A002457.

Cf. A000290 (row sums), A002457,, A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.

[(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1,n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022

gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
seq(seq(T(n,k), k=0..n),n=0..7); # Peter Luschny, Jan 18 2020
S:=(n,k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
rho:=n->[seq(S(n,k),k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od: # N. J. A. Sloane, Jan 18 2020

Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k,k]*Binomial[n+k,n-k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)

flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k,k)*binomial(n+k+1,n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022

T(n, 0) = (-1)^n*A000027(n+1).
T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).
T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).
T(n, n-2) = A002738(n-2).
T(n, n-1) = (-1)*A002736(n).
T(n, n) = A002457(n).
T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).
Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).
Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).
2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).
(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).
T(n, k) = ((2*k+1)!/(k!)^2)*[x^(n-k)] (1+x)^(-2*(k+1)). - Georg Fischer and Peter Luschny, Jan 18 2020
T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - N. J. A. Sloane, Jan 18 2020
From Petros Hadjicostas, Jul 09 2020: (Start)
Michael Somos's formulas above can be restated as
Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.
These are special cases of the following formula that is alluded to (in some way) in Ser's book:
1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).
Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)
T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022

Several typos in the data corrected by Georg Fischer and Peter Luschny, Jan 18 2020

Definition changed by N. J. A. Sloane, Jan 18 2020

A103194 LAH transform of squares.

Original entry on oeis.org

0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977, 589410910, 9064804551, 149641946796, 2638693215769, 49490245341642, 983607047803815, 20646947498718736, 456392479671188001, 10595402429677269174, 257723100178182605287, 6553958557721713088820

Offset: 0

Vladeta Jovovic, Mar 18 2005

If the e.g.f. of b(n) is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*b(k), then the e.g.f. of a(n) is E(x/(1-x))/(1-x). - Vladeta Jovovic, Apr 16 2005
a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are in a cycle.  A fixed point is considered to be in a cycle.  a(n) = Sum_{k=0..n} A206703(n,k)*k. - Geoffrey Critzer, Feb 11 2012
a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are undefined, i.e., they do not have an image.- Geoffrey Critzer, Feb 09 2022
a(n) is the total length of all increasing subsequences over all n-permutations.  Cf. A002720. - Geoffrey Critzer, Feb 09 2022

Alois P. Heinz, Table of n, a(n) for n = 0..200
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.
N. J. A. Sloane, Transforms

Cf. A000262, A000290, A001477, A206703.

with(combstruct): SetSeqSetL := [T, {T=Set(S), S=Sequence(U, card >= 1), U=Set(Z, card=1)}, labeled]: seq(k*count(SetSeqSetL, size=k), k=0..18); # Zerinvary Lajos, Jun 06 2007
a := n -> n!*hypergeom([2, 1-n], [1, 1], -1):
seq(simplify(a(n)),n=0..20); # Peter Luschny, Mar 30 2015

nn = 20; a = 1/(1 - x); ay = 1/(1 - y x); D[Range[0, nn]! CoefficientList[ Series[Exp[a x] ay, {x, 0, nn}], x], y] /. y -> 1  (* Geoffrey Critzer, Feb 11 2012 *)

a(n) = Sum_{k=0..n} (n!/k!)*binomial(n-1, k-1)*k^2.
E.g.f.: x/(1-x)^2*exp(x/(1-x)).
Recurrence: (n-1)*a(n) - n*(2*n-1)*a(n-1) + n*(n-1)^2*a(n-2) = 0.
a(n) = n*A000262(n). - Vladeta Jovovic, Mar 20 2005
a(n) ~ n! * exp(-1/2 + 2*sqrt(n))*n^(1/4)/(2*sqrt(Pi)). - Vaclav Kotesovec, Aug 13 2013
a(n) = n!*hypergeom([2, 1-n], [1, 1], -1). - Peter Luschny, Mar 30 2015
a(n) = Sum_{k=1..n} k*(n-k)!*binomial(n,k)^2. - Ridouane Oudra, Jun 17 2025

A270228 Number of matchings in the n X n rook graph K_n X K_n.

Original entry on oeis.org

1, 7, 370, 270529, 3337807996, 855404716021831, 5352265402523357926168, 940288991338542314571521981185, 5236753179470435264288904589157765055760, 1029720447530443779943631183186535523331685533812231

Offset: 1

Andrew Howroyd, Mar 13 2016

nonn

K_n X K_n is also called the rook graph or lattice graph.

Andrew Howroyd, Table of n, a(n) for n = 1..16
Andrew Howroyd, C# software related to this sequence
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Rook Graph
Wikipedia, Rook's graph, Hosoya index

Cf. A270227, A270229, A085537 (Wiener index), A002720 (independent vertex sets), A269561, A028420.

A002695 P_n'(3), where P_n is n-th Legendre polynomial.

Original entry on oeis.org

1, 9, 66, 450, 2955, 18963, 119812, 748548, 4637205, 28537245, 174683718, 1064611782, 6464582943, 39132819495, 236256182280, 1423046656008, 8554078990377, 51327262010673, 307488810131530, 1839455028693450

Offset: 1

N. J. A. Sloane, Simon Plouffe

H. Bateman, Some problems in potential theory, Messenger Math., 52 (1922), 71-78.
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=1..100
H. Bateman, Some problems in potential theory, Messenger Math., 52 (1922), 71-78. [Annotated scanned copy]
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.

Cf. A001850.

Table[SeriesCoefficient[x*(1-6x+x^2)^(-3/2),{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 04 2012 *)
a[n_]:= Sum[(i Binomial[n+i+1,i] Binomial[n+1,i]),{i,1,n+1}]/2
Table[a[n], {n, 0, 20}] (* Gerry Martens, Apr 08 2018 *)

N = 66;  x = 'x + O('x^N);
gf = x*(1-6*x+x^2)^(-3/2);
Vec(gf)
/* Joerg Arndt, Mar 29 2013 */

G.f.: x*(1-6*x+x^2)^(-3/2). [corrected by Vaclav Kotesovec, Oct 04 2012]
a(n) = Gegenbauer_C(n,3/2,3). - Paul Barry, Apr 20 2009
D-finite with recurrence: -n*a(n-2) + 3*(2*n-1)*a(n-1) + (1-n)*a(n) = 0. - Vaclav Kotesovec, Oct 04 2012
a(n) ~ (3+2*sqrt(2))^n*sqrt(n)/(4*sqrt(2*Pi)*sqrt(3*sqrt(2)-4)). - Vaclav Kotesovec, Oct 04 2012
a(n) = (n+1) * n * A001003(n)/2, n>0. - Vladimir Kruchinin, Mar 29 2013
a(n) = Sum_{i=1..n} i*binomial(n+i,i)*binomial(n,i)/2. - Gerry Martens, Apr 08 2018

A021010 Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).

Original entry on oeis.org

1, -1, 1, 1, -4, 2, -1, 9, -18, 6, 1, -16, 72, -96, 24, -1, 25, -200, 600, -600, 120, 1, -36, 450, -2400, 5400, -4320, 720, -1, 49, -882, 7350, -29400, 52920, -35280, 5040, 1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320, -1, 81, -2592, 42336, -381024, 1905120, -5080320, 6531840, -3265920, 362880

Offset: 0

N. J. A. Sloane

abs(T(n,k)) = k!*binomial(n,k)^2 = number of k-matchings of the complete bipartite graph K_{n,n}. Example: abs(T(2,2))=2 because in the bipartite graph K_{2,2} with vertex sets {A,B},{A',B'} we have the 2-matchings {AA',BB'} and {AB',BA'}. Row sums of the absolute values yield A002720. - Emeric Deutsch, Dec 25 2004

			   1;
  -1,   1;
   1,  -4,   2;
  -1,   9, -18,   6;
   1, -16,  72, -96,  24;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

T. D. Noe, Rows n = 0..50 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].
J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
C. Lanczos, Applied Analysis (Annotated scans of selected pages) See page 519.
Eric Weisstein's World of Mathematics, Rook Polynomial
Kin Yip Wong, A Dynamic Coupling Model of Optical Conductivity in Mixed-Valence Systems, arXiv:2410.13144 [cond-mat.mtrl-sci], 2024. See p. 12.
Index entries for sequences related to Laguerre polynomials

Cf. A002720, A021009, A009940 (row sums).

Central terms: A295383.

[[(-1)^(n-k)*Factorial(k)*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

T:=(n,k)->(-1)^(n-k)*k!*binomial(n,k)^2: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, Dec 25 2004

Flatten[ Table[ Reverse[ CoefficientList[n!*LaguerreL[n, x], x]], {n, 0, 9}]] (* Jean-François Alcover, Nov 24 2011 *)

LaguerreL(n,v='x) = {
  my(x='x+O('x^(n+1)), t='t);
  subst(polcoeff(exp(-x*t/(1-x))/(1-x), n), 't, v);
};
concat(apply(n->Vec(n!*LaguerreL(n)), [0..9])) \\ Gheorghe Coserea, Oct 26 2017

row(n) = Vec(n!*pollaguerre(n)); \\ Michel Marcus, Feb 06 2021

T(n, k) = (-1)^(n-k)*k!*binomial(n, k)^2. - Emeric Deutsch, Dec 25 2004

A023997 Number of block permutations on an n-set.

Original entry on oeis.org

1, 1, 3, 25, 339, 6721, 179643, 6166105, 262308819, 13471274401, 818288740923, 57836113793305, 4693153430067699, 432360767273547841, 44794795522199781243, 5176959027946049635225, 662704551840482536170579

Offset: 0

Des FitzGerald (D.FitzGerald(AT)utas.edu.au)

A block permutation of a set X is a bijection between two quotient sets of X (of necessarily equal rank).

Number of labeled partitions of (n,n) into pairs (i,j) where there are n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object. - Christian G. Bower, Jun 03 2005

			For n=3, there are the 3! ordinary permutations (of rank 3), 18 block permutations of rank 2 (2! for each pair of partitions of rank 2) and the single rank 1 one.

Vaclav Kotesovec, Table of n, a(n) for n = 0..288 (terms 0..45 from Vincenzo Librandi)
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 22.
D. G. FitzGerald and Jonathan Leech, Dual symmetric inverse monoids and representation theory, J. Australian Mathematical Society (Series A), Vol. 64 (1998), pp. 345-367.

Cf. A023998, A002720, A014235, A111420.

Table[Sum[StirlingS2[n,k]^2k!,{k,0,n}],{n,0,100}] (* Emanuele Munarini, Jul 04 2011 *)

makelist(sum(stirling2(n,k)^2*k!,k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */

a(n) = if (n==0, 1, sum(k=1, n, k!*stirling(n, k, 2)^2)); \\ Michel Marcus, Jun 18 2019

a(0)=1, a(n) = Sum_{k=1..n} k! * S2(n,k)^2, S2(n,k) are the Stirling numbers of the second kind.

More terms from Christian G. Bower, Jun 03 2005

A288268 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*x^k/k).

Original entry on oeis.org

1, 0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080

Offset: 0

Seiichi Manyama, Oct 20 2017

Seiichi Manyama, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A000296, A002720, A008275, A009940, A052852, A052887, A059114, A288269.

l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
[1,0]cat[(Factorial(n)/(n-1))*(2*l(n-1,0,-1) - l(n,0,-1)): n in [2..30]]; // G. C. Greubel, Mar 10 2021

a := proc(n) option remember; if n < 3 then [1, 0, 1][n+1] else
-(n^2 - 4*n + 3)*a(n - 2) + (2*n - 2)*a(n - 1) fi end:
seq(a(n), n = 0..21); # Peter Luschny, Feb 20 2022

Table[If[n<2, 1-n, (n!/(n-1))*(2*LaguerreL[n-1, -1] - LaguerreL[n, -1])], {n, 0, 30}] (* G. C. Greubel, Mar 10 2021 *)

{a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)*x^k/k)+x*O(x^n)), n)}

[1-n if n<2 else (factorial(n)/(n-1))*(2*gen_laguerre(n-1,0,-1) - gen_laguerre(n,0,-1)) for n in (0..30)] # G. C. Greubel, Mar 10 2021

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)*a(n-k)/(n-k)! for n > 0.
E.g.f.: (1 - x) * exp(x/(1 - x)). - Ilya Gutkovskiy, Jul 27 2020
a(n) = (n!/(n-1))*( 2*LaguerreL(n-1, -1) - LaguerreL(n, -1) ) with a(0) = 1, a(1) = 0. - G. C. Greubel, Mar 10 2021
a(n) ~ n^(n - 3/4) * exp(-1/2 + 2*sqrt(n) - n) / sqrt(2) * (1 - 65/(48*sqrt(n))). - Vaclav Kotesovec, Mar 10 2021, minor term corrected Dec 01 2021
From Peter Luschny, Feb 20 2022: (Start)
a(n) = n! * Sum_{k=0..n} (-1)^k * LaguerreL(n-k, k-1, -1).
a(n) = 2*(n - 1)*a(n - 1) - (n^2 - 4*n + 3)*a(n - 2) for n >= 3. (End)
From Peter Bala, May 26 2023: (Start)
a(n) = Sum_{k = 0..n} |Stirling1(n,k)|*A000296(k) (follows from the fundamental theorem of Riordan arrays).
Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is purely periodic with the period dividing k. For example, modulo 7 we obtain the purely periodic sequence [1, 0, 1, 4, 0, 3, 2, 1, 0, 1, 4, 0, 3, 2, ...] of period 7. Cf. A047974. (End)
For n>1, a(n) = (2*n*A002720(n-1) - A002720(n))/(n-1). - Vaclav Kotesovec, May 27 2023

A002793 a(n) = 2na(n-1) - (n-1)^2a(n-2).

Original entry on oeis.org

0, 1, 4, 20, 124, 920, 7940, 78040, 859580, 10477880, 139931620, 2030707640, 31805257340, 534514790680, 9591325648580, 182974870484120, 3697147584561340, 78861451031150840, 1770536585183202980, 41729280102868841080, 1030007496863617367420, 26568602827124392999640

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

nonn

From Wolfdieter Lang, Dec 12 2011: (Start)
r(n) = a(n+1)*(-1)^n, n >= 0, gives the alternating row sums of the coefficient triangle A199577, i.e., r(n)=La_n(1;0,-1), with the monic first associated Laguerre polynomials with parameter alpha=0 evaluated at x=-1.
The e.g.f. for these row sums r(n) is g(x) = -(2+x)*exp(1/(1+x))*(Ei(1,1/(1+x))-Ei(1,1))/(1+x)^3 + 1/(1+x)^2, with the exponential integral Ei(1,x) = Gamma(0,x).
This e.g.f. satisfies the homogeneous ordinary second-order differential equation (1+x)^2*(d^2/dx^2)g(x) + (6+5*x)*(d/dx)g(x) + 4*g(x) = 0, g(0)=1, (d/dx)g(x)|_{x=0}=-4.
This e.g.f. g(x) is equivalent to the recurrence
  b(n)= -2*(n+1)*b(n-1) - n^2*b(n-2), b(-1)=0, b(0)=1.
Therefore, the e.g.f. of a(n) is A(x)=int(g(-x),x), with A(0)=0. This agrees with the e.g.f. given below in the formula section by Max Alekseyev.
(End)

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
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).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.

G. C. Greubel, Table of n, a(n) for n = 0..440
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

Bisection of A056952. A199577 (alternating row sums, unsigned).

Cf. A002720, A073003.

I:=[1, 4]; [0] cat [n le 2 select I[n] else 2*n*Self(n-1) - (n-1)^2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018

Flatten[{0,RecurrenceTable[{(-1+n)^2 a[-2+n]-2 n a[-1+n]+a[n]==0,a[1]==1,a[2]==4}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)
nxt[{n_,a_,b_}]:={n+1,b,2(n+1)b-n^2 a}; NestList[nxt,{1,0,1},30][[All,2]] (* Harvey P. Dale, Sep 06 2022 *)

A058006(n) = sum(k=0,n, (-1)^k*k! );
a(n) = if (n<=1, n, sum(k=1, n, (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)! ) ); /* Joerg Arndt, Oct 12 2012 */

{a(n)=if(n==1,1,polcoeff(1-sum(m=1, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} \\ Paul D. Hanna, Feb 06 2013

From Max Alekseyev, Jul 06 2010: (Start)
For n > 1, a(n) = Sum_{k=1..n} (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)!.
E.g.f.: (Gamma(0,1) - Gamma(0,1/(1-x))) * exp(1/(1-x)) / (1-x). (End)
From Peter Bala, Oct 11 2012: (Start)
Numerators in the sequence of convergents of Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := int {x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The denominators are in A002720.
(End)
G.f.: x = Sum_{n>=1} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Feb 06 2013
a(n) ~ G * exp(2*sqrt(n) - n - 1/2) * n^(n+1/4) / sqrt(2) * (1 + 31/(48*sqrt(n))), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013

Edited by Max Alekseyev, Jul 13 2010

cp's OEIS Frontend

A277421 a(n) = n!*LaguerreL(n, -7*n).

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A277422 a(n) = n!*LaguerreL(n, -8*n).

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A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

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A103194 LAH transform of squares.

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A270228 Number of matchings in the n X n rook graph K_n X K_n.

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A002695 P_n'(3), where P_n is n-th Legendre polynomial.

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A021010 Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).

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A277421 a(n) = n!LaguerreL(n, -7n).

A277422 a(n) = n!LaguerreL(n, -8n).

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

A002793 a(n) = 2na(n-1) - (n-1)^2a(n-2).