A002720 -id:A002720 - cp's OEIS frontend

A160617 Numerator of Laguerre(n, -1).

1, 2, 7, 17, 209, 773, 13327, 65461, 1441729, 1255151, 234662231, 1702678841, 53334454417, 448162154317, 16083557845279, 13946689584823, 126523856174033, 66120494322107921, 269906478537389909, 34987413853951524577

Offset: 0

N. J. A. Sloane, Nov 14 2009

G. C. Greubel, Table of n, a(n) for n = 0..440

For denominators see A160618.

Cf. A002720.

[Numerator((&+[Binomial(n,k)*(1/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 06 2018

Numerator[Table[LaguerreL[n, -1], {n, 0, 50}]] (* G. C. Greubel, May 06 2018 *)

s(n):=if n=0 then 1/2 else 1/(n+2)*(n +1 +sum(s(n-i-2)*(i+1), i,0,n-2));
makelist(num(s(n)),n,0,20); /* Vladimir Kruchinin, Sep 30 2016 */

lista(nn) = {x = 'x + O('x^nn); v = exp(x/(1-x))/(1-x); for (n=0, nn-1, print1(numerator(polcoeff(v, n)), ", "););} \\ Michel Marcus, Nov 27 2015

for(n=0,30, print1(numerator(sum(k=0,n, binomial(n,k)*(1/k!))), ", ")) \\ G. C. Greubel, May 06 2018

a(n) = numerator(s(n)), where s(0)=1/2, s(n) = 1/(n+2)*(n +1 + Sum_{i=0..n-2} s(n-i-2)*(i+1)). - Vladimir Kruchinin, Sep 30 2016

A160618 Denominator of Laguerre(n, -1).

Original entry on oeis.org

1, 1, 2, 3, 24, 60, 720, 2520, 40320, 25920, 3628800, 19958400, 479001600, 3113510400, 87178291200, 59439744000, 426995712000, 177843714048000, 582033973248000, 60822550204416000, 143111882833920000

Offset: 0

N. J. A. Sloane, Nov 14 2009

G. C. Greubel, Table of n, a(n) for n = 0..449

For numerators see A160617.

Cf. A002720.

[Denominator((&+[Binomial(n,k)*(1/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 06 2018

Denominator[Table[LaguerreL[n, -1], {n, 0, 50}]] (* G. C. Greubel, May 06 2018 *)

lista(nn) = {x = 'x + O('x^nn); v = exp(x/(1-x))/(1-x); for (n=0, nn-1, print1(denominator(polcoeff(v, n)), ", "););} \\ Michel Marcus, Nov 27 2015

for(n=0,30, print1(denominator(sum(k=0,n, binomial(n,k)*(1/k!))), ", ")) \\ G. C. Greubel, May 06 2018

A182925 Generalized vertical Bell numbers of order 3.

Original entry on oeis.org

1, 15, 1657, 513559, 326922081, 363303011071, 637056434385865, 1644720885001919607, 5943555582476814384769, 28924444943026683877502191, 183866199607767992029159792281, 1489437787210535537087417039489815

Offset: 0

Peter Luschny, Mar 28 2011

nonn

The name "generalized 'vertical' Bell numbers" is used to distinguish them from the generalized (horizontal) Bell numbers with reference to the square array representation of the generalized Bell numbers as given in A090210. a(n) is column 4 in this representation. The order is the parameter M in Penson et al., p. 6, eq. 29.

G. C. Greubel, Table of n, a(n) for n = 0..168
P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation, (2010).
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp,
Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Cf. A090210, A002720, A069948, A182924.

A182925 := proc(n) exp(-x)*GAMMA(n+1)^3*hypergeom([n+1,n+1,n+1],[1,1,1],x);
round(evalf(subs(x=1,%),64)) end; seq(A182925(i),i=0..11);

u = 1.`64; a[n_] := n!^3*HypergeometricPFQ[{n+u, n+u, n+u}, {u, u, u}, u]/E // Round; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Nov 22 2012, after Maple *)

a(n) = exp(-1)*Gamma(n+1)^3*[3F3]([n+1, n+1, n+1], [1, 1, 1] | 1); here [3F3] is the generalized hypergeometric function of type 3F3.

Let B_{n}(x) = Sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 4! [x^4] taylor(B_{n}(x)), where [x^4] denotes the coefficient of x^4 in the Taylor series for B_{n}(x).

A283500 Triangle read by rows: T(n,k) = number of n X n (0,1) matrices with at most k 1's in each row or column.

Original entry on oeis.org

2, 7, 16, 34, 265, 512, 209, 7343, 41503, 65536, 1546, 304186, 6474726, 24997921, 33554432, 13327, 17525812, 1709852332, 21252557377, 57366997447, 68719476736, 130922, 1336221251, 702998475376, 34215495252681, 252540841305558, 505874809287625

Offset: 1

R. J. Mathar, Mar 09 2017

			Triangle begins:
2;
7,     16;
34,    265,      512;
209,   7343,     41503,      65536;
1546,  304186,   6474726,    24997921,    33554432;
13327, 17525812, 1709852332, 21252557377, 57366997447, 68719476736;
...

Andrew Howroyd, Table of n, a(n) for n = 1..84

Cf. A002720 (column k=1), A197458 (column k=2), A008300 (exactly k 1s).
Main diagonal and first lower diagonal give: A002416, A048291.
Cf. A247158 (k=n/2).

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

A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 15, 5, 9, 13, 20, 17, 27, 37, 52, 6, 11, 16, 25, 21, 34, 47, 67, 26, 43, 60, 87, 77, 114, 151, 203, 7, 13, 19, 30, 25, 41, 57, 82, 31, 52, 73, 107, 94, 141, 188, 255, 37, 63, 89, 132, 115, 175, 235, 322, 141, 218, 295, 409, 372, 523, 674

Offset: 0

Mikhail Kurkov, Aug 23 2021 [verification needed]

Modulo 2 binomial transform of A243499(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Kevin Ryde, PARI/GP Code

Cf. A000110, A000120, A002720, A005493, A005494, A007814, A008277, A035009, A045379, A053645, A129760, A143494, A143495, A196834, A265649, A295989.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

function a = A347204(max_n)
    a(1) = 1;
    a(2) = 2;
    for nloop = 3:max_n
        n = nloop-1;
        s = 0;
        for k = 0:floor(log2(n))-1
            s = s + a(1+A053645(n)-2^k*(mod(floor(n/(2^k)),2)));
        end
        a(nloop) = 2*a(A053645(n)+1) + s;
    end
end
function a_n = A053645(n)
    a_n = n - 2^floor(log2(n));
end % Thomas Scheuerle, Oct 25 2021

f[n_] := BitAnd[n, n - 1]; a[0] = 1; a[n_] := a[n] = a[f[n]/2] + a[Floor[(n + f[n])/2]]; Array[a, 100, 0] (* Amiram Eldar, Nov 19 2021 *)

f(n) = bitand(n, n-1); \\ A129760
a(n) = if (n<=1, n+1, if (n%2, a(n\2)+a(n-1), a(f(n/2)) + a(n/2+f(n/2)))); \\ Michel Marcus, Oct 25 2021

PARI
```
\\ Also see links.
```

A129760(n) = bitand(n, n-1);
memoA347204 = Map();
A347204(n) = if (n<=1, n+1, my(v); if(mapisdefined(memoA347204,n,&v), v, v = if(n%2, A347204(n\2)+A347204(n-1), A347204(A129760(n/2)) + A347204(n/2+A129760(n/2))); mapput(memoA347204,n,v); (v))); \\ (Memoized version of Michel Marcus's program given above) - Antti Karttunen, Nov 20 2021

a(n) = a(n - 2^f(n)) + (1 + f(n))*a((n - 2^f(n))/2) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(2n+1) = a(n) + a(2n) for n >= 0.
a(2n) = a(n - 2^f(n)) + a(2n - 2^f(n)) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) - 2^k*T(n,k)) for n > 1 with a(0) = 1, a(1) = 2, and where f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2.
Sum_{k=0..2^n - 1} a(k) = A035009(n+1) for n >= 0.
a((4^n - 1)/3) = A002720(n) for n >= 0.
a(2^n - 1) = A000110(n+1),
a(2*(2^n - 1)) = A005493(n),
a(2^2*(2^n - 1)) = A005494(n),
a(2^3*(2^n - 1)) = A045379(n),
a(2^4*(2^n - 1)) = A196834(n),
a(2^m*(2^n-1)) = T(n,m+1) is the n-th (m+1)-Bell number for n >= 0, m >= 0 where T(n,m) = m*T(n-1,m) + Sum_{k=0..n-1} binomial(n-1,k)*T(k,m) with T(0,m) = 1.
a(n) = Sum_{j=0..2^A000120(n)-1} A243499(A295989(n,j)) for n >= 0. Also A243499(n) = Sum_{j=0..2^f(n)-1} (-1)^(f(n)-f(j)) a(A295989(n,j)) for n >= 0 where f(n) = A000120(n). In other words, a(n) = Sum_{j=0..n} (binomial(n,j) mod 2)*A243499(j) and A243499(n) = Sum_{j=0..n} (-1)^(f(n)-f(j))*(binomial(n,j) mod 2)*a(j) for n >= 0 where f(n) = A000120(n).
Generalization:
b(n, x) = (1/x)*b((n - 2^f(n))/2, x) + (-1)^n*b(floor((2n - 2^f(n))/2), x) for n > 0 with b(0, x) = 1 where f(n) = A007814(n).
Sum_{k=0..2^n - 1} b(k, x) = (1/x)^n for n >= 0.
b((4^n - 1)/3, x) = (1/x)^n*n!*L_{n}(x) for n >= 0 where L_{n}(x) is the n-th Laguerre polynomial.
b((8^n - 1)/7, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A265649(n, k) for n >= 0.
b(2^n - 1, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A008277(n+1, k+1),
b(2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143494(n+2, k+2),
b(2^2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143495(n+3, k+3),
b(2^m*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*T(n+m+1, k+m+1, m+1) for n >= 0, m >= 0 where T(n,k,m) is m-Stirling numbers of the second kind.

A361595 Expansion of e.g.f. exp( (x / (1-x))^3 ) / (1-x).

Original entry on oeis.org

1, 1, 2, 12, 120, 1320, 15480, 199080, 2862720, 46146240, 826156800, 16212873600, 344741443200, 7875365097600, 192137321376000, 4984375210214400, 136994756496998400, 3976455027389644800, 121533921410994892800, 3900447928934548992000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A002720, A361594.

Cf. A361572.

Table[n! * Sum[Binomial[n,3*k]/k!, {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^3)/(1-x)))

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/k!.
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (4*n - 3)*a(n-1) - 3*(n-1)*(2*n - 3)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) - (n-3)^2*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(-1/8) * exp(-1/4 + 5*3^(-1/4)*n^(1/4)/8 - sqrt(3*n)/2 + 4*3^(-3/4) * n^(3/4) - n) * n^(n + 1/8) / 2 * (1 + (1511/2560)*3^(1/4)/n^(1/4)). (End)

A361596 Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).

Original entry on oeis.org

1, 1, 3, 15, 99, 795, 7485, 80745, 981225, 13253625, 196834995, 3185662095, 55770765435, 1049572599075, 21120725230605, 452384160453225, 10272547048388625, 246434674107647025, 6226347228582355875, 165224032352989584975, 4593512876411509125075

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A002720, A361597.

Cf. A335344.

Table[n! * Sum[Binomial[n,2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

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

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n,2*k)/(2^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (3*n - 2)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).
a(n) ~ 3^(-1/2) * exp(1/6 - n^(1/3)/2 + 3*n^(2/3)/2 - n) * n^(n + 1/6) * (1 + 49/(108*n^(1/3)) + 3293/(116640*n^(2/3))). (End)

A361597 Expansion of e.g.f. exp( x^3/(6 * (1-x)^3) ) / (1-x).

Original entry on oeis.org

1, 1, 2, 7, 40, 320, 3130, 34930, 432320, 5866840, 86816800, 1395455600, 24270908200, 454897042600, 9146979842000, 196443726879400, 4486709145318400, 108548344109004800, 2771885136281060800, 74475606190225240000, 2099591224223100608000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A002720, A361596.

Cf. A361573.

Table[n! * Sum[Binomial[n,3*k]/(6^k * k!), {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3))/(1-x)))

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/(6^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
Recurrence: 2*a(n) = 2*(4*n - 3)*a(n-1) - 6*(n-1)*(2*n - 3)*a(n-2) + (n-2)*(n-1)*(8*n - 17)*a(n-3) - 2*(n-3)^2*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(-7/8) * exp(-1/24 + 5*2^(-15/4)*n^(1/4)/3 - sqrt(n/2)/2 + 2^(7/4)*n^(3/4)/3 - n) * n^(n + 1/8) * (1 + (2637/10240)*2^(3/4)/n^(1/4)). (End)

A002738 Coefficients for extrapolation.

Original entry on oeis.org

3, 60, 630, 5040, 34650, 216216, 1261260, 7001280, 37413090, 193993800, 981608628, 4867480800, 23728968900, 114011377200, 540972351000, 2538963567360, 11802213457650, 54396360988200, 248812984520100, 1130341536324000, 5103492036502860, 22913637714910800

Offset: 0

N. J. A. Sloane

nonn

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-2 of B equals a(n-3). - T. D. Noe, May 01 2011

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
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)

Cf. A002802, A007744, A020918.

A diagonal of A331431.

[3*Binomial(2*n+3,n)*Binomial(n+3,3): n in [0..30]]; // G. C. Greubel, Mar 21 2022

Table[Total[Inverse[HilbertMatrix[n]][[n - 2]]], {n, 3, 25}] (* T. D. Noe, May 02 2011 *)

[3*binomial(2*n+3,3)*binomial(2*n,n) for n in (0..30)] # G. C. Greubel, Mar 21 2022

From Alois P. Heinz, May 02 2011: (Start)
a(n) = 3*binomial(2*n+3,n)*binomial(n+3,n).
G.f.: 3*(1 + 6*x)/(1-4*x)^(7/2). (End)
a(n) = binomial(2*n+3,n)*(n^3 + 6*n^2 + 11*n+6)/2. - Charles R Greathouse IV, May 02 2011
a(n) = 3*A007744(n). - R. J. Mathar, Jan 21 2020
a(n) = (3/2)*( 5*A020918(n) - 3*A002802(n)). - G. C. Greubel, Mar 21 2022

Extended by T. D. Noe, May 01 2011

cp's OEIS Frontend

A160617 Numerator of Laguerre(n, -1).

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A160618 Denominator of Laguerre(n, -1).

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A182925 Generalized vertical Bell numbers of order 3.

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A283500 Triangle read by rows: T(n,k) = number of n X n (0,1) matrices with at most k 1's in each row or column.

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

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Magma

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A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

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A361595 Expansion of e.g.f. exp( (x / (1-x))^3 ) / (1-x).

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Mathematica

PARI

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A361596 Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).

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