A102773 -id:A102773 - cp's OEIS frontend

A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

1, 5, 34, 286, 2840, 32344, 414160, 5876336, 91356544, 1542401920, 28075364096, 547643910400, 11389266525184, 251428006132736, 5869482147358720, 144413021660821504, 3733822274973040640, 101181690628832198656, 2867011297057247002624, 84764595415605494743040

Offset: 0

Alois P. Heinz, Jun 26 2017

nonn

Number of marriage patterns between a labeled set X of n women and a labeled set Y of n men (all heterosexual): some couples can be formed where one partner is from X and the other from Y, some members of X and Y marry external (unlabeled) partners, and some do not marry.

			a(1) = 5:
[0 0]  [1 0]  [0 1]  [0 0]  [0 1]
[0 0]  [0 0]  [0 0]  [1 0]  [1 0] .
.
a(2) = 34:
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]
.
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[0 0 1]  [0 1 0]  [0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]
[1 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]
[1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]
[0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [1 0 0]
.
[0 1 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]
[1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 1 0]
[0 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]  [0 0 0]  .

Alois P. Heinz, Table of n, a(n) for n = 0..437
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=4 of A289192.

Cf.: A000142, A000165, A000302, A002720, A025167, A084771, A087912, A102773, A160611, A160612, A277382.

a:= proc(n) option remember; `if`(n<2, 4*n+1,
      (2*n+3)*a(n-1)-(n-1)^2*a(n-2))
    end:
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n-> n! * add(binomial(n, i)*4^i/i!, i=0..n):
seq(a(n), n=0..25);
# third Maple program:
a:= n-> n!* simplify(LaguerreL(n, -4), 'LaguerreL'):
seq(a(n), n=0..25);

Table[n! LaguerreL[n, -4], {n, 0, 30}] (* Indranil Ghosh, Jul 06 2017 *)

from mpmath import *
mp.dps=150
l=chop(taylor(lambda x:exp(4*x/(1-x))/(1-x), 0, 31))
print([int(fac(i)*l[i]) for i in range(len(l))]) # Indranil Ghosh, Jul 06 2017
# or #
from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -4))
print([a(n) for n in range(31)]) # Indranil Ghosh, Jul 06 2017

E.g.f.: exp(4*x/(1-x))/(1-x).
a(n) = Sum_{i=0..n} i! * (2^(n-i)*binomial(n,i))^2.
a(n) = Sum_{i=0..n} (n-i)! * 4^i * binomial(n,i)^2.
a(n) = n! * Sum_{i=0..n} 4^i/i! * binomial(n,i).
a(n) = (2*n+3)*a(n-1)-(n-1)^2*a(n-2) for n>=2, a(n) = 4*n+1 for n<2.
a(n) = n! * Laguerre(n,-4) = n! * A160611(n)/A160612(n).
a(n) ~ exp(-2 + 4*sqrt(n) - n) * n^(n + 1/4) / 2 * (1 + 163/(96*sqrt(n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 4^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A341014 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * j! * binomial(n,j)^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 17, 34, 1, 1, 5, 31, 139, 209, 1, 1, 6, 49, 352, 1473, 1546, 1, 1, 7, 71, 709, 5233, 19091, 13327, 1, 1, 8, 97, 1246, 13505, 95836, 291793, 130922, 1, 1, 9, 127, 1999, 28881, 318181, 2080999, 5129307, 1441729, 1

Offset: 0

Seiichi Manyama, Feb 02 2021

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     3,     4,      5,      6, ...
  1,    7,    17,    31,     49,     71, ...
  1,   34,   139,   352,    709,   1246, ...
  1,  209,  1473,  5233,  13505,  28881, ...
  1, 1546, 19091, 95836, 318181, 830126, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 0..4 give A000012, A002720, A025167, A102757, A102773.
Rows 0..2 give A000012, A000027(n+1), A056220(n+1).
Main diagonal gives A330260.
Cf. A307883.

T[n_, k_] := Sum[If[j == k == 0, 1, k^j]*j!*Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 02 2021 *)

{T(n,k) = sum(j=0, n, k^j*j!*binomial(n, j)^2)}

E.g.f. of column k: exp(x/(1-k*x)) / (1-k*x).

T(n,k) = (2*k*n-k+1) * T(n-1,k) - k^2 * (n-1)^2 * T(n-2,k) for n > 1.

A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 2, 17, 352, 13505, 830126, 74717857, 9263893892, 1513712421377, 315230799073690, 81499084718806001, 25612081645835777192, 9615370149488574778177, 4250194195208050117007942, 2184834047906975645398282625, 1292386053018890618812398220876

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A002720, A025167, A061711, A102757, A102773, A187021, A277373, A277452, A293146, A330497.

Main diagonal of A341014.

[Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019

Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ Michel Marcus, Dec 18 2019

a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselI(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

A102757 a(n) = Sum_{i=0..n} C(n,i)^2 * i! * 3^i.

Original entry on oeis.org

1, 4, 31, 352, 5233, 95836, 2080999, 52189096, 1482977857, 47053929268, 1648037039791, 63125834205424, 2624096058047281, 117620219281363852, 5653607876781921463, 290035426344483253816, 15814774125898034896129

Offset: 0

Miklos Kristof, Mar 16 2005

Primes in this sequence include: a(2)=31, a(4)=5233. Semiprimes in this sequence include: a(1) = 2^2, a(6) = 31 * 67129, a(8) = 127 * 11676991. - Jonathan Vos Post, Mar 17 2005

Seiichi Manyama, Table of n, a(n) for n = 0..377

Cf. A002720, A025167, A102773.

seq(sum('binomial(k,i)^2*i!*3^i', 'i'=0..k),k=0..30);

f[n_] := Sum[k!*3^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* or *)
Range[0, 16]! CoefficientList[ Series[1/(1 - 3x)*Exp[x/(1 - 3x)], {x, 0, 16}], x] (* Robert G. Wilson v, Mar 16 2005 *)

E.g.f.: 1/(1-3x)*exp(x/(1-3x)).
E.g.f.: exp(3*x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [Paul D. Hanna, Nov 18 2011]
a(n) = 2*(3*n-1)*a(n-1) - 9*(n-1)^2*a(n-2). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ (3*n)^(n+1/4)*exp(2*sqrt(n/3)-n-1/6)/sqrt(2) * (1 + 103/(144*sqrt(3*n))). - Vaclav Kotesovec, Sep 29 2013

More terms from Robert G. Wilson v, Mar 16 2005

A121079 a(n) = Sum_{i=0..n} C(n,i)^2i!4^i + 2^n*n!.

Original entry on oeis.org

2, 7, 57, 757, 13889, 322021, 8962225, 289928549, 10666353409, 439225736005, 19999574572721, 997265831223685, 54028099173536449, 3159178743189436709, 198259676112757095985, 13289233274778582230821, 947420482287986880154625, 71574264415491967142194309

Offset: 0

N. J. A. Sloane, Aug 11 2006

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..363
Joël Gay, Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
Z. Li, Z. Li and Y. Cao, Enumeration of symplectic and orthogonal injective partial transformations, Discrete Math., 306 (2006), 1781-1787.

Cf. A102773, A121080.

Array[Sum[Binomial[#, i]^2*i!*4^i, {i, 0, #}] + 2^#*#! &, 18, 0] (* Michael De Vlieger, Nov 28 2018 *)

a(n) = 2^n*n! + sum(i=0, n, binomial(n,i)^2*i!*4^i); \\ Michel Marcus, May 31 2018

A121080 a(n) = Sum_{i=0..n} C(n,i)^2i!4^i + (1-2^n)2^(n-1)n!.

Original entry on oeis.org

1, 4, 37, 541, 10625, 258661, 7464625, 248318309, 9339986689, 391569431365, 18095180332721, 913513359466885, 50008961524486849, 2950209091316054309, 186558089772409191985, 12587159519294553302821, 902488447534988078746625, 68518909362619336345906309

Offset: 0

N. J. A. Sloane, Aug 11 2006

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..363
Joël Gay, Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
Z. Li, Z. Li and Y. Cao, Enumeration of symplectic and orthogonal injective partial transformations, Discrete Math., 306 (2006), 1781-1787.

Cf. A102773, A121079.

Array[Sum[Binomial[#, i]^2*i!*4^i, {i, 0, #}] + (1 - 2^#)*2^(# - 1)*#! &, 18, 0] (* Michael De Vlieger, Nov 28 2018 *)

a(n) = (1-2^n)*2^(n-1)*n! + sum(i=0, n, binomial(n,i)^2*i!*4^i); \\ Michel Marcus, May 31 2018

cp's OEIS Frontend

A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

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A341014 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * j! * binomial(n,j)^2.

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A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

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A102757 a(n) = Sum_{i=0..n} C(n,i)^2 * i! * 3^i.

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A121079 a(n) = Sum_{i=0..n} C(n,i)^2*i!*4^i + 2^n*n!.

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A121080 a(n) = Sum_{i=0..n} C(n,i)^2*i!*4^i + (1-2^n)*2^(n-1)*n!.

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A121079 a(n) = Sum_{i=0..n} C(n,i)^2i!4^i + 2^n*n!.

A121080 a(n) = Sum_{i=0..n} C(n,i)^2i!4^i + (1-2^n)2^(n-1)n!.