A087981 -id:A087981 - cp's OEIS frontend

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

1, 0, 2, 6, 72, 620, 8640, 122346, 2156672, 41367672, 905126400, 21646532270, 570077595648, 16268377195044, 502096929431552, 16629319748711250, 588938142209310720, 22196966267762213744, 887352465220427317248, 37496112562144553167062, 1670071417348195942400000, 78195398849926292810318940

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

The n-th term of the n-fold exponential convolution of A000166 with themselves.

Robert Israel, Table of n, a(n) for n = 0..396
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Main diagonal of A295181.

Cf. A000166, A000407, A001813, A087981, A137775, A295183, A383379.

S:= series((exp(-x)/(1-x))^n,x,30):
seq(n!*coeff(S,x,n),n=0..29); # Robert Israel, Nov 16 2017

Table[n! SeriesCoefficient[Exp[-n x]/(1 - x)^n, {x, 0, n}], {n, 0, 21}]

a(n) = A295181(n,n).
a(n) ~ phi^(3*n - 1/2) * n^n / (5^(1/4) * exp(n*(1 + 1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017
a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k-1,k)/(n-k)!. - Seiichi Manyama, Apr 25 2025

A383344 Expansion of e.g.f. exp(-4*x) / (1-x)^4.

Original entry on oeis.org

1, 0, 4, 8, 72, 416, 3520, 31104, 316288, 3525632, 43117056, 572195840, 8191304704, 125761056768, 2060841582592, 35894401335296, 662066514984960, 12890305925218304, 264155723747688448, 5682905054074109952, 128051031032232411136, 3015653024970577018880

Offset: 0

Seiichi Manyama, Apr 23 2025

Column k=4 of A295181.
Cf. A000166, A087981, A137775.
Cf. A088991, A381504.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-4*x)/(1-x)^4))

a(n) = n! * Sum_{k=0..n} (-4)^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) = (n-1) * (a(n-1) + 4*a(n-2)) for n > 1.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A000166.
a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+4)). - Vaclav Kotesovec, Apr 25 2025

A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

Original entry on oeis.org

1, 3, 14, 82, 576, 4688, 43264, 445632, 5062016, 62812288, 844863744, 12239474432, 189939644416, 3142842052608, 55223903596544, 1026805938614272, 20139224002953216, 415503046091767808, 8994794537935765504, 203848794955954716672, 4826475681472562855936, 119162892472107134353408

Offset: 0

Seiichi Manyama, Apr 24 2025

Cf. A001909, A383382, A383383, A383384.

Cf. A000023, A052124, A087981, A383380.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^5))

a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = 1, a(1) = 3; a(n) = (n+2)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A087982 Maximal permanent of a nonsingular n X n (+1,-1)-matrix.

Original entry on oeis.org

1, 0, 2, 8, 24, 128

Offset: 1

N. J. A. Sloane, Oct 28 2003

nonn

It is conjectured by Kraeuter and Seifter that for n >= 5 the maximal permanent of a nonsingular n X n (+1,-1)-matrix is attained by a matrix with exactly n-1 -1's on the diagonal (compare A087981).
This has been proved by Budrevich and Guterman. - Sergei Shteiner, Jan 21 2020
The maximal possible value for the permanent of a singular n X n (+1,-1)-matrix is obviously n!.

			a(4) = 8 from the following matrix:
-1 +1 +1 +1
+1 +1 +1 +1
+1 -1 +1 -1
-1 +1 +1 -1

Mikhail V. Budrevich, Alexander E. Guterman, Kräuter conjecture on permanents is true, arXiv:1810.04439 [math.CO], 2018.
Arnold R. Kräuter and Norbert Seifter, Some properties of the permanent of (1,-1)-matrices, Linear and Multilinear Algebra 15 (1984), 207-223.
Norbert Seifter, Upper bounds for permanents of (1,-1)-matrices, Israel J. Math. 48 (1984), 69-78.
Edward Tzu-Hsia Wang, On permanents of (1,-1)-matrices, Israel J. Math. 18 (1974), 353-361.
Index entries for sequences related to binary matrices

For n != 4 this is given by A087981. Cf. A087983.

a(n) = A087981(n-1) for n >= 5. - Sergei Shteiner, Jan 20 2020

a(4) = 8 from W. Edwin Clark and Wouter Meeussen, a(5) = 24 and a(6) = 128 from Jaap Spies, Oct 29 2003

A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

Original entry on oeis.org

0, 1, -2, 6, -8, 40, 48, 784, 5248, 49536, 490240, 5403904, 64822272, 842742784, 11798284288, 176974510080, 2831591636992, 48137058942976, 866467058614272, 16462874118651904, 329257482362552320, 6914407129635618816, 152116956851937476608, 3498690007594658430976

Offset: 0

Ilya Gutkovskiy, May 23 2020

sign

Inverse binomial transform of A000240.

Cf. A000023, A000240, A008290, A066534, A087981, A122803.

Table[n! Sum[(-2)^k/k!, {k, 0, n - 1}], {n, 0, 23}]
nmax = 23; CoefficientList[Series[Sum[k! x^k/(1 + 2 x)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[x Exp[-2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n! * sum(k=0, n-1, (-2)^k / k!); \\ Michel Marcus, May 23 2020

G.f.: Sum_{k>=1} k! * x^k / (1 + 2*x)^(k + 1).
E.g.f.: x*exp(-2*x) / (1 - x).
a(n) = A000023(n) - A122803(n).
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Jun 08 2022
a(n) = Sum_{k=0..n} (-1)^k * k * A008290(n,k). - Alois P. Heinz, May 20 2023

A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

Original entry on oeis.org

1, 2, 8, 40, 248, 1808, 15136, 142784, 1496960, 17254144, 216740864, 2945973248, 43065951232, 673626675200, 11224114860032, 198447384666112, 3710328985124864, 73136238041563136, 1515739708283944960, 32947698735175172096, 749499782353468522496, 17806903161183314378752

Offset: 0

Seiichi Manyama, Apr 24 2025

Cf. A000261, A383344, A383378.
Cf. A000023, A052124, A087981, A383381.
Cf. A000255.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^4))

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A000255.
a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+3,3)/(n-k)!.
a(0) = 1, a(1) = 2; a(n) = (n+1)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(Pi) * n^(n + 7/2) / (3*sqrt(2)*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A176097 Degree of the hyperdeterminant of the cubic format (k+1) X (k+1) X (k+1).

Original entry on oeis.org

1, 4, 36, 272, 2150, 16992, 134848, 1072192, 8536914, 68036600, 542607560, 4329671040, 34561892560, 275979195520, 2204266118400, 17609217372416, 140698273234634, 1124340854572296, 8985828520591912, 71822662173752800

Offset: 0

Benjamin J. Young, Apr 08 2010

			For k=1, the hyperdeterminant of the matrix (a_ijk) (for 0 <= i,j,k <= 1) is (a_000 * a_111)^2 + (a001 * a110)^2 + (a_010 * a_101)^2 + (a_011 * a_100)^2 -2(a_000 * a_001 * a_110 * a_111 + a_000 * a_010 * a_101 * a_111 + a_000 * a_011 * a_100 * a_111 + a_001 * a_010 * a_101 * a_110 + a_001 * a_011 * a_110 * a_100 + a_010 * a_011 * a_101 * a_100) + 4(a_000 * a_011 * a_101 * a_110 + a_001 * a_010 * a_100 * a_111) (see Gelfand, Kapranov & Zelevinsky, pp. 2 and 448.) [Corrected by _Petros Hadjicostas_, Sep 12 2019]

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 2008, p. 456 (Ch. 14, Corollary 2.9).

Arthur Cayley, On the theory of linear transformations, The Cambridge Mathematical Journal, Vol. IV, No. XXIII, February 1845, pp. 193-209. [Accessible only in the USA through the Hathi Trust Digital Library.]
Arthur Cayley, On the theory of linear transformations, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 80-94. [Accessible through the University of Michigan Historical Math Collection; click on pp. 80 through 94.]
Arthur Cayley, On linear transformations, Cambridge and Dublin Mathematical Journal, Vol. I, 1846, pp. 104-122. [Accessible only in the USA through the Hathi Trust Digital Library.]
Arthur Cayley, On linear transformations, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 95-112. [Accessible through the University of Michigan Historical Math Collection; click on pp. 95 through 112.]
I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Hyperdeterminants, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.9 (p. 246).
David G. Glynn, The modular counterparts of Cayley's hyperdeterminants, Bulletin of the Australian Mathematical Society 57(3) (1998), 479-492.
Giorgio Ottaviani, Luca Sodomaco, and Emuanuele Ventura, Asymptotics of degrees and ED degrees of Segre products, arXiv:2008.11670 [math.AG], 2020.
Ludwig Schläfli, Über die Resultante eines Systemes mehrerer algebraischen Gleichungen, ein Beitrag zur Theorie der Elimination, Denkschr. der Kaiserlicher Akad. der Wiss. math-naturwiss. Klasse, 4 Band, 1852.
Eric Weisstein's World of Mathematics, Hyperdeterminant.
Wikipedia, Hyperdeterminants.

Cf. A045899, A050269, A086302, A087981.

a:= k-> add((j+k+1)! /(j!)^3 /(k-2*j)! *2^(k-2*j), j=0..floor(k/2)): seq(a(n), n=0..20);
# Second program:
a := proc(n) option remember; if n = 0 then return 1 elif n = 1 then return 4 fi;
(a(n-1)*(21*n^3-10*n^2-9*n+6)+a(n-2)*(24*n^3+16*n^2))/((3*n-1)*n^2) end:
seq(a(n), n=0..19); # Peter Luschny, Sep 12 2019

Table[Sum[(j + n + 1)!*2^(n - 2*j)/(j!^3*(n - 2*j)!), {j, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 12 2019 *)

a(n) = Sum_{j = 0..n/2} ( (j+n+1)! * 2^(n-2j) )/((j!)^3 * (n-2j)!).
a(n) = (n+1)^2*(8*A000172(n)-A000172(n+1))/6. - Mark van Hoeij, Jul 02 2010
G.f.: hypergeom([-1/3, 1/3],[1],27*x^2/(1-2*x)^3)*(1-2*x)/((x+1)^2*(1-8*x)). - Mark van Hoeij, Apr 11 2014
a(n) ~ 8^(n+1) / (Pi * 3^(3/2)). - Vaclav Kotesovec, Sep 12 2019
a(n) = (a(n-1)*(21*n^3 - 10*n^2 - 9*n + 6) + a(n-2)*(24*n^3 + 16*n^2))/((3*n - 1)*n^2) for n >= 2. - Peter Luschny, Sep 12 2019

A335595 E.g.f.: exp(-x * (2 + x)) / (1 - x)^2.

Original entry on oeis.org

1, 0, 0, 4, 12, 48, 400, 3120, 25872, 251776, 2715264, 31809600, 405296320, 5580385536, 82469607168, 1302102360832, 21875297337600, 389590168842240, 7331376554610688, 145352459953603584, 3028176414606560256, 66135374473635328000, 1510938930307368898560, 36038691473858577444864

Offset: 0

Ilya Gutkovskiy, Nov 20 2020

nonn

Cf. A038205, A087981.

A335595 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        2*add(binomial(n-1,k-1)*(k-1)!*procname(n-k),k=3..n) ;
    end if;
end proc:
seq(A335595(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

nmax = 23; CoefficientList[Series[Exp[-x (2 + x)]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] HermiteH[k, -1] (n - k + 1)!, {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(-x*(2+x))/(1-x)^2)) \\ Michel Marcus, Nov 21 2020

a(0) = 1; a(n) = 2 * Sum_{k=3..n} binomial(n-1,k-1) * (k-1)! * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * Hermite(k,-1) * (n-k+1)!.
a(n) = Sum_{k=0..n} binomial(n,k) * A038205(k) * A038205(n-k).
a(n) ~ exp(-3) * n * n!. - Vaclav Kotesovec, Aug 09 2021
D-finite with recurrence a(n) +(-n+1)*a(n-1) -2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Aug 20 2021

A176901 Number of 3 X n semireduced Latin rectangles, that is, having exactly n fixed points in the first two rows.

Original entry on oeis.org

4, 72, 1584, 70720, 3948480, 284570496, 25574768128, 2808243910656, 369925183388160, 57585548812887040, 10458478438093154304, 2191805683821733404672, 525011528578874444283904, 142540766765931981615759360, 43542026550306796238178877440, 14867182204795857282384287236096, 5640920219495105293649671985430528

Offset: 3

Vladimir Shevelev, Apr 28 2010

nonn

A Latin rectangle is called reduced if its first row is [1,2,...,n] (the number of 3 X n reduced Latin rectangles is given in A000186). Therefore a Latin rectangle having exactly n fixed points in the first two rows may be called "semireduced". Thus if A1(i), A2(i), i=1,...,n, are the first two rows, then, for every i, either A1(i)=i or A2(i)=i.

V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr. Mat.(J. of the Akademy of Sciences of Russia) 4(1992), 91-110.
V. S. Shevelev, Modern enumeration theory of permutations with restricted positions, Diskr. Mat., 1993, 5, no.1, 3-35 (Russian).
V. S. Shevelev, Modern enumeration theory of permutations with restricted positions, English translation, Discrete Math. and Appl., 1993, 3:3, 229-263 (pp. 255-257).

Cf. A174563, A000179, A000186, A087981, A094047, A174556, A174560, A174561.

Let F_n = A087981(n) = n! * Sum_{2*k_2+...+n*k_n=n, k_i>=0} Product_{i=2..n} 2^k_i/(k_i!*i^k_i). Then a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * F_k * F_(n-k) * u_(n-2*k), where u(n) = A000179(n). - Vladimir Shevelev, Mar 30 2016

More terms from William P. Orrick, Jul 25 2020

A337615 T(n, k) = binomial(n, k)sf(n-k)sf(k) where sf is the subfactorial (A000166). Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 2, 0, 0, 2, 9, 0, 6, 0, 9, 44, 0, 20, 20, 0, 44, 265, 0, 135, 80, 135, 0, 265, 1854, 0, 924, 630, 630, 924, 0, 1854, 14833, 0, 7420, 4928, 5670, 4928, 7420, 0, 14833, 133496, 0, 66744, 44520, 49896, 49896, 44520, 66744, 0, 133496

Offset: 0

Peter Luschny, Sep 05 2020

			Triangle starts:
[0]      1;
[1]      0, 0;
[2]      1, 0,     1;
[3]      2, 0,     0,     2;
[4]      9, 0,     6,     0,     9;
[5]     44, 0,    20,    20,     0,    44;
[6]    265, 0,   135,    80,   135,     0,   265;
[7]   1854, 0,   924,   630,   630,   924,     0,  1854;
[8]  14833, 0,  7420,  4928,  5670,  4928,  7420,     0, 14833;
[9] 133496, 0, 66744, 44520, 49896, 49896, 44520, 66744,     0, 133496.

Cf. A000166 (T(n,0) and T(n,n)), A087981 (row sums).

sf := n -> add((-1)^(n-j)*pochhammer(n-j+1, j), j=0..n):
T := (n, k) -> binomial(n,k)*sf(n-k)*sf(k):
seq(seq(T(n, k), k=0..n), n=0..9);

cp's OEIS Frontend

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

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A383344 Expansion of e.g.f. exp(-4*x) / (1-x)^4.

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A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

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A087982 Maximal permanent of a nonsingular n X n (+1,-1)-matrix.

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A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

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A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

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A176097 Degree of the hyperdeterminant of the cubic format (k+1) X (k+1) X (k+1).

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A335595 E.g.f.: exp(-x * (2 + x)) / (1 - x)^2.

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A176901 Number of 3 X n semireduced Latin rectangles, that is, having exactly n fixed points in the first two rows.

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A337615 T(n, k) = binomial(n, k)sf(n-k)sf(k) where sf is the subfactorial (A000166). Triangle read by rows, for 0 <= k <= n.