A277382 -id:A277382 - 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

A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)n^(n-k)n!/(n-k)!.

Original entry on oeis.org

0, 1, 10, 141, 2584, 58745, 1602576, 51165205, 1874935168, 77644293201, 3588075308800, 183111507687581, 10230243235200000, 621111794820235849, 40722033570202507264, 2867494972696071121125, 215840579093024990396416, 17294837586403146090259745, 1469799445329208661211021312

Offset: 0

Peter Luschny, Oct 11 2016

nonn

Cf. A097662, A239768.

Cf. A002720, A087912, A277382.

a := n -> add(binomial(n,n-k)*n^(n-k)*n!/(n-k)!, k=1..n):
seq(a(n), n=0..18);
# Alternatively:
A277372 := n -> n!*LaguerreL(n,-n) - n^n:
seq(simplify(A277372(n)), n=0..18);

a(n) = sum(k=1, n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!); \\ Michel Marcus, Oct 12 2016

a(n) = n!*LaguerreL(n, -n) - n^n.
a(n) = (-1)^n*KummerU(-n, 1, -n) - n^n.
a(n) = n^n*(hypergeom([-n, -n], [], 1/n) - 1) for n>=1.
a(n) ~ n^n * phi^(2*n+1) * exp(n/phi-n) / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2016

A160613 Numerator of Laguerre(n, -3).

Original entry on oeis.org

1, 4, 23, 28, 491, 1249, 19223, 15476, 3515161, 1512661, 14496817, 228800107, 11770539419, 60428965661, 5262254717509, 2521163372543, 976843770850217, 887131806309703, 73511154681979031, 255777165814872577

Offset: 0

N. J. A. Sloane, Nov 14 2009

G. C. Greubel, Table of n, a(n) for n = 0..480

For denominators see A160614.

Cf. A277382.

[Numerator((&+[Binomial(n,k)*(3^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018

Numerator[Table[LaguerreL[n, -3], {n, 0, 50}]] (* G. C. Greubel, May 09 2018 *)

for(n=0,30, print1(numerator(sum(k=0,n, binomial(n,k)*(3^k/k!))), ", ")) \\ G. C. Greubel, May 09 2018

A160614 Denominator of Laguerre(n, -3).

Original entry on oeis.org

1, 1, 2, 1, 8, 10, 80, 35, 4480, 1120, 6400, 61600, 1971200, 6406400, 358758400, 112112000, 28700672000, 17425408000, 975822848000, 2317579264000, 370812682240000, 49917091840000, 57105153064960000, 41044328765440000

Offset: 0

N. J. A. Sloane, Nov 14 2009

G. C. Greubel, Table of n, a(n) for n = 0..499

For numerators see A160613.

Cf. A277382.

[Denominator((&+[Binomial(n,k)*(3^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 06 2018

Denominator[Table[LaguerreL[n, -3], {n, 0, 50}]] (* G. C. Greubel, May 06 2018 *)

for(n=0,30, print1(denominator(sum(k=0,n, binomial(n,k)*(3^k/k!))), ", ")) \\ G. C. Greubel, May 06 2018

A277386 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 3^(n-k) * k!.

Original entry on oeis.org

1, 4, 35, 438, 6873, 127488, 2703447, 64121130, 1674999009, 47638235484, 1461975938379, 48068355965886, 1683311251028265, 62477888170824792, 2447583053876363727, 100842325515959413842, 4356021203508275392833, 196739133595421931988020, 9268144156277932321747251

Offset: 0

Vaclav Kotesovec, Oct 12 2016

nonn

Cf. A000172, A277382, A241247, A274246.

Table[Sum[Binomial[n, k]^3 * 3^(n-k) * k!, {k, 0, n}], {n, 0, 20}]

Recurrence: n*(8*n - 23)*a(n) = 3*(8*n^3 - 15*n^2 - 30*n + 17)*a(n-1) - (n-1)*(24*n^3 - 261*n^2 + 770*n - 666)*a(n-2) + (n-2)^3*(n-1)*(8*n - 15)*a(n-3).
a(n) ~ n^(n - 1/6) * exp(3*3^(1/3)*n^(2/3) - 3^(2/3)*n^(1/3) - n +1) / (3^(5/6)*sqrt(2*Pi)) * (1 + 19/(6*3^(2/3)*n^(1/3)) + 1193/(1080*3^(1/3) * n^(2/3))).
Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} 3^n * x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022

A343847 T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 7, 3, 1, 24, 34, 14, 4, 1, 120, 209, 86, 23, 5, 1, 720, 1546, 648, 168, 34, 6, 1, 5040, 13327, 5752, 1473, 286, 47, 7, 1, 40320, 130922, 58576, 14988, 2840, 446, 62, 8, 1, 362880, 1441729, 671568, 173007, 32344, 4929, 654, 79, 9, 1

Offset: 0

Peter Luschny, May 07 2021

			Triangle starts:
0:     1;
1:     1,      1;
2:     2,      2,     1;
3:     6,      7,     3,     1;
4:    24,     34,    14,     4,    1;
5:   120,    209,    86,    23,    5,   1;
6:   720,   1546,   648,   168,   34,   6,  1;
7:  5040,  13327,  5752,  1473,  286,  47,  7,  1;
8: 40320, 130922, 58576, 14988, 2840, 446, 62,  8,  1;
.
Array whose upward read antidiagonals are the rows of the triangle.
n\k   0       1       2        3        4         5        6
-----------------------------------------------------------------
0:    1,      1,      1,       1,       1,        1,        1, ...
1:    1,      2,      3,       4,       5,        6,        7, ...
2:    2,      7,     14,      23,      34,       47,       62, ...
3:    6,     34,     86,     168,     286,      446,      654, ...
4:   24,    209,    648,    1473,    2840,     4929,     7944, ...
5:  120,   1546,   5752,   14988,   32344,    61870,   108696, ...
6:  720,  13327,  58576,  173007,  414160,   866695,  1649232, ...
7: 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, ...

Row sums: A343848. T(2*n, n) = A277373(n). Variant: A289192.
Columns: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Cf. A021009 (Laguerre polynomials), A344048.

T := proc(n, k) option remember;
if n = k then return 1 elif n = k+1 then return k+1 fi;
(2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) end:
seq(print(seq(T(n ,k), k = 0..n)), n = 0..7);

T[n_, k_] := (-1)^(n - k) HypergeometricU[k - n, 1, -k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Alternative: *)
TL[n_, k_] := (n - k)! LaguerreL[n - k, -k];
Table[TL[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

T(n, k) = (n - k)!*sum(j=0, n - k, binomial(n - k, j) * k^j / j!)
for(n=0, 9, for(k=0, n, print(T(n, k))))

# Columns of the array.
def column(k, len):
    R. = PowerSeriesRing(QQ, default_prec=len)
    f = exp(k * x / (1 - x)) / (1 - x)
    return f.egf_to_ogf().list()
for col in (0..6): print(column(col, 20))

T(n, k) = (-1)^(n - k)*U(k - n, 1, -k), where U is the Kummer U function.
T(n, k) = (n - k)! * L(n - k, -k), where L is the Laguerre polynomial function.
T(n, k) = (n - k)! * Sum_{j = 0..n - k} binomial(n - k, j) k^j / j!.
T(n, k) = (2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) for n - k >= 2.

A299018 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P(n) = n(x + 1)P(n - 1) - (n - 2)^2xP(n - 2).

Original entry on oeis.org

1, 2, 2, 6, 11, 6, 24, 60, 60, 24, 120, 366, 501, 366, 120, 720, 2532, 4242, 4242, 2532, 720, 5040, 19764, 38268, 46863, 38268, 19764, 5040, 40320, 172512, 373104, 528336, 528336, 373104, 172512, 40320, 362880, 1668528, 3942108, 6237828, 7213761, 6237828, 3942108, 1668528, 362880

Offset: 1

F. Chapoton, Jan 31 2018

			For n = 3, the polynomial is 6*x^2 + 11*x + 6.
The first few polynomials, as a table:
[1],
[2,    2],
[6,    11,    6],
[24,   60,    60,  24],
[120,  366,   501, 366, 120]

Very similar to A298854.
Row sums are A277382(n-1) for n>0.
Leftmost and rightmost columns are A000142.
Alternating row sums are A177145.
Alternating row sum of row 2*n+1 is A001818(n).

P:= proc(n) option remember; expand(`if`(n<2, n,
      n*(x+1)*P(n-1)-(n-2)^2*x*P(n-2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(P(n)):
seq(T(n), n=1..12);  # Alois P. Heinz, Jan 31 2018
A := proc(n,k) ## n >= 0 and k = 0 .. n
    option remember;
    if n = 0 and k = 0 then
        1
    elif n > 0 and k >= 0 and k <= n then
        (n+1)*(A(n-1,k)+A(n-1,k-1))-(n-1)^2*A(n-2,k-1)
    else
        0
    end if;
end proc: # Yu-Sheng Chang, Apr 14 2020

P[n_] := P[n] = Expand[If[n < 2, n, n (x+1) P[n-1] - (n-2)^2 x P[n-2]]];
row[n_] := CoefficientList[P[n], x];
row /@ Range[12] // Flatten (* Jean-François Alcover, Dec 10 2019 *)

@cached_function
def poly(n):
    x = polygen(ZZ, 'x')
    if n < 1:
        return x.parent().zero()
    elif n == 1:
        return x.parent().one()
    else:
        return n * (x + 1) * poly(n - 1) - (n - 2)**2 * x * poly(n - 2)

P(0) = 0, P(1) = 1 and P(n) = n * (x + 1) * P(n - 1) - (n - 2)^2 * x * P(n - 2).

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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A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.

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A160613 Numerator of Laguerre(n, -3).

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A160614 Denominator of Laguerre(n, -3).

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A277386 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 3^(n-k) * k!.

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A343847 T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.

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A299018 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P(n) = n*(x + 1)*P(n - 1) - (n - 2)^2*x*P(n - 2).

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A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)n^(n-k)n!/(n-k)!.

A299018 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P(n) = n(x + 1)P(n - 1) - (n - 2)^2xP(n - 2).