A009940 -id:A009940 - cp's OEIS frontend

A317365 Expansion of e.g.f. x*exp(x/(1 + x))/(1 + x).

0, 1, 0, -3, 16, -75, 336, -1295, 1632, 55881, -1124000, 16722981, -229985040, 3089923837, -41225160144, 545880027225, -7069180940864, 86130735547665, -882387869940288, 3847692639294541, 171852333163131600, -8392137456287472699, 276055495385982856720, -8067943451470397940543

Offset: 0

Ilya Gutkovskiy, Jul 26 2018

sign

Inverse Lah transform of the nonnegative integers (A001477).

G. C. Greubel, Table of n, a(n) for n = 0..440
N. J. A. Sloane, Transforms
Index entries for sequences related to Laguerre polynomials

Cf. A001477, A009940, A052852.

[n eq 0 select 0 else (-1)^(n+1)*Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), 1): n in [0..25]]; // G. C. Greubel, Mar 05 2021

a:= proc(n) option remember; add((-1)^(n-k)*
      n!/(k-1)!*binomial(n-1, k-1), k=1..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

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

a(n) = (-1)^(n+1)*n!*pollaguerre(n-1,0,1); \\ Michel Marcus, Mar 06 2021

[0 if n==0 else (-1)^(n+1)*factorial(n)*gen_laguerre(n-1, 0, 1) for n in (0..25)] # G. C. Greubel, Mar 05 2021

a(n) = Sum_{k=1..n} binomial(n-1,k-1)*n!/(k-1)!.
From G. C. Greubel, Mar 05 2021: (Start)
a(n) = n! * Hypergeometric1F1([-(n-1)], [1], -1).
a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1). (End)

A331325 a(n) = n!*[x^n] cosh(x/(1-x))/(1-x).

Original entry on oeis.org

1, 1, 3, 15, 97, 745, 6571, 65359, 723969, 8842257, 118091251, 1712261551, 26786070433, 449634481465, 8059974923547, 153634497337455, 3102367733191681, 66145005096272929, 1484586887025099619, 34983117545622446287, 863397428225495045601, 22269844592814969946761

Offset: 0

Peter Luschny, Jan 21 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..443

Cf. A002720, A009940, A021009, A331326.

gf := cosh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):
seq(n!*coeff(ser, x, n), n=0..21);
# Alternative: seq(add(abs(A021009(n, 2*k)), k=0..n/2), n=0..21);
A331325 := proc(n) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*add(S(n, 2*k), k=0..n) end:
seq(A331325(n), n=0..21);

a[n_] := n! HypergeometricPFQ[{1/2 - n/2, -n/2}, {1, 1/2, 1/2}, 1/4];
Array[a, 22, 0]

x='x+O('x^22); Vec(serlaplace(cosh(x/(1-x))/(1-x)))

def A331325():
    sa, sb, ta, tb, n = 1, 2, 1, 0, 2
    yield sa
    yield ta
    while(True):
        s = 2*n*sb - ((n-1)**2)*sa
        t = 2*(n-1)*tb - ((n-1)**2)*ta
        sa, sb, ta, tb = sb, s, tb, t
        n += 1
        yield (s + t)//2
a = A331325(); print([next(a) for _ in range(22)])

a(n) + A331326(n) = A002720(n).
a(n) - A331326(n) = A009940(n).
a(n) = Sum_{k=0..n/2} |A021009(n, 2*k)|.
a(n) = Sum_{k=0..n} binomial(n, 2*k)*n!/(2*k)!.
a(n) = n!*hypergeom([1/2 - n/2, -n/2], [1/2, 1/2, 1], 1/4).
(n+1)^2*(n+2)^2*a(n) - 4*(n+2)^3*a(n+1) + (6*n^2+30*n+37)*a(n+2) - 4*(n+3)*a(n+3)+a(n+4)=0. - Robert Israel, Jan 23 2020
Sum_{n>=0} a(n) * x^n / (n!)^2 = (1/2) * exp(x) * (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))). - Ilya Gutkovskiy, Jul 18 2020
a(n) ~ 2^(-3/2) * exp(2*sqrt(n)-n-1/2) * n^(n+1/4) * (1 + 31/(48*sqrt(n))). - Vaclav Kotesovec, Feb 17 2024

A331326 a(n) = n!*[x^n] sinh(x/(1 - x))/(1 - x).

Original entry on oeis.org

0, 1, 4, 19, 112, 801, 6756, 65563, 717760, 8729857, 116570980, 1693096131, 26548383984, 446689827169, 8023582921732, 153192673528651, 3097301219335936, 66095983547942913, 1484384376886189380, 34991710162280602867, 863797053818651591920, 22282392569877969167521

Offset: 0

Peter Luschny, Jan 21 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..443

Cf. A002720, A009940, A021009, A331325.

gf := sinh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):
seq(n!*coeff(ser, x, n), n=0..20);
# Alternative: seq(add(abs(A021009(n, 2*k+1)), k=0..n/2), n=0..21);
A331326 := proc(n) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*add(S(n, 2*k+1), k=0..n) end:
seq(A331326(n), n=0..21);

a[n_] := n n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1, 3/2, 3/2}, 1/4];
Array[a, 22, 0]

x='x+O('x^22); concat(0,Vec(serlaplace(sinh(x/(1-x))/(1-x))))

def A331326():
    sa, sb, ta, tb, n = 1, 2, 1, 0, 2
    yield 0
    yield ta
    while(True):
        s = 2*n*sb - ((n-1)**2)*sa
        t = 2*(n-1)*tb - ((n-1)**2)*ta
        sa, sb, ta, tb = sb, s, tb, t
        n += 1
        yield (s - t)//2
a = A331326(); print([next(a) for _ in range(22)])

a(n) + A331325(n) = A002720(n).
A331325(n) - a(n) = A009940(n).
a(n) = Sum_{k=0..n/2} |A021009(n, 2*k+1)|.
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*n!/(2*k+1)!.
a(n) = n*n!*hypergeom([1/2 - n/2, 1 - n/2], [1, 3/2, 3/2], 1/4).
(n+1)^2*(n+2)^2*a(n) - 4*(n+2)^3*a(n+1) + (6*n^2+30*n+37)*a(n+2) - 4*(n+3)*a(n+3)+a(n+4) = 0. - Robert Israel, Jan 22 2020
Sum_{n>=0} a(n) * x^n / (n!)^2 = (1/2) * exp(x) * (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))). - Ilya Gutkovskiy, Jul 17 2020
a(n) ~ 2^(-3/2) * exp(2*sqrt(n)-n-1/2) * n^(n+1/4) * (1 + 31/(48*sqrt(n))). - Vaclav Kotesovec, Feb 17 2024

A331688 E.g.f.: exp(-x/(1 - x)) / (1 - 2*x).

Original entry on oeis.org

1, 1, 3, 17, 137, 1389, 16819, 236557, 3792753, 68326073, 1366917731, 30074632521, 721798881913, 18766625660197, 525460685327187, 15763716503597189, 504436925448024929, 17150818356045629937, 617428780939911647683, 23462281235407345160833

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..403

Cf. A000165, A000166, A000354, A009940, A293116, A331689.

f:= gfun:-rectoproc({a(n) = -(n - 1)*(5*n - 8)*a(n - 2) + (-3 + 4*n)*a(n - 1) + 2*(n - 1)*(n - 2)^2*a(n - 3),a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 28 2020

nmax = 19; CoefficientList[Series[Exp[-x/(1 - x)]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! Subfactorial[n - k], {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n,k)^2 * k! * A000166(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * k! * 2^k * A293116(n-k).
a(n) ~ n! * exp(-1) * 2^n. - Vaclav Kotesovec, Jan 26 2020
a(n) = (4*n-3)*a(n-1)-(n-1)*(5*n-8)*a(n-2)+2*(n-1)*(n--2)^2*a(n-3). - Robert Israel, Jul 28 2020

A331725 E.g.f.: exp(x/(1 - x)) / (1 + x).

Original entry on oeis.org

1, 0, 3, 4, 57, 216, 2755, 18348, 247569, 2368432, 35256771, 436248660, 7235178313, 108919083144, 2010150360387, 35421547781116, 723689454172065, 14543895730321248, 326843345169621379, 7354350135365751972, 180610925178770615001, 4488323611011676811320

Offset: 0

Ilya Gutkovskiy, Jan 25 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A002720, A009940, A133942, A182386, A206703.

A331725 := proc(n)
    add((-1)^k*binomial(n,k)*k!*A000262(n-k),k=0..n) ;
end proc:
seq(A331725(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

nmax = 21; CoefficientList[Series[Exp[x/(1 - x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
A000262[n_] := If[n == 0, 1, n! Sum[Binomial[n - 1, k]/(k + 1)!, {k, 0, n - 1}]]; a[n_] := Sum[(-1)^k Binomial[n, k] k! A000262[n - k], {k, 0, n}]; Table[a[n], {n, 0, 21}]
a[n_] := (-1)^n n! (1 - Sum[(-1)^j*LaguerreL[j, 1, -1]/(j+1), {j,0,n-1}]);
Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 20 2022 *)

seq(n)={Vec(serlaplace(exp(x/(1 - x) + O(x*x^n)) / (1 + x)))} \\ Andrew Howroyd, Jan 25 2020

def gen_a():
    F, L, S, N = 1, 1, 1, 1
    while True:
        yield F * S
        L = gen_laguerre(N - 1, 1, -1) / N
        S += L if F < 0 else -L
        F *= -N; N += 1
a = gen_a(); print([next(a) for  in range(21)]) # _Peter Luschny, Feb 20 2022

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * k! * A000262(n-k).
a(n) ~ n^(n - 1/4) / (2^(3/2) * exp(1/2 - 2*sqrt(n) + n)). - Vaclav Kotesovec, Jan 26 2020
D-finite with recurrence a(n) +(-n+1)*a(n-1) -(n-1)*(n+1)*a(n-2) +(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Aug 20 2021
a(n) = Sum_{k=0..n} (-1)^k * A206703(n,k). - Alois P. Heinz, Feb 19 2022
a(n) = (-1)^n*n!*(1 - Sum_{j=0..n-1}((-1)^j*LaguerreL(j, 1, -1)/(j + 1))). - Peter Luschny, Feb 20 2022

A336292 a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(n-k) / (k * ((n-k)!)^2).

Original entry on oeis.org

0, 1, -2, 3, 8, 305, 10734, 502747, 30344992, 2307890097, 216571514030, 24619605092291, 3337294343698248, 532148381719443073, 98646472269855762238, 21041945289232131607995, 5118447176652195630775424, 1408601897794844346184122017, 435481794298015565250651718302

Offset: 0

Ilya Gutkovskiy, Jul 16 2020

sign

Cf. A002741, A009940, A336291.

Table[(n!)^2 Sum[(-1)^(n - k)/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[1 - x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * sum(k=1, n, (-1)^(n-k) / (k * ((n-k)!)^2)); \\ Michel Marcus, Jul 17 2020

Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(1 - x) * BesselJ(0,2*sqrt(x)).

A087860 Expansion of e.g.f.: (1-exp(x/(x-1)))/(1-x).

Original entry on oeis.org

0, 1, 3, 10, 39, 176, 905, 5244, 34111, 250480, 2108529, 20751380, 241315151, 3282366504, 50786289385, 865850559196, 15856276032255, 306665879765984, 6199863566817761, 130237717066988580, 2832527601333186319

Offset: 0

Vladeta Jovovic, Oct 25 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..448

Cf. A009940, A070779.

I:=[1,3,10]; [0] cat [n le 3 select I[n] else 3*(n-1)*Self(n-1) - (n-1)*(3*n-5)*Self(n-2) +(n-1)*(n-2)^2*Self(n-3): n in [1..30]];

With[{nn=20},CoefficientList[Series[(1-Exp[x/(x-1)])/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 27 2015 *)

x='x+O('x^30); concat([0], Vec(serlaplace((1-exp(x/(x-1)))/(1-x)))) \\ G. C. Greubel, Feb 06 2018

a(n) = n!*(1 - LaguerreL(n, 1)).
a(n) = 3*(n-1)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * (1 - BesselJ(0,2*sqrt(x))). - Ilya Gutkovskiy, Jul 17 2020
a(n) = n*n!*hypergeom([1 - n, 1], [2, 2], 1). - Peter Luschny, May 10 2021
a(n) ~ n! * (1 - exp(1/2)*cos(2*sqrt(n) - Pi/4) / (sqrt(Pi) * n^(1/4))). - Vaclav Kotesovec, May 10 2021

Definition clarified by Harvey P. Dale, Nov 27 2015

A253667 Square array read by ascending antidiagonals, T(n, k) = k![x^k](exp(-x) sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, -1, -1, 1, 2, -1, 2, 1, 1, 3, 1, -1, -3, -1, 1, 4, 5, -4, 5, 4, 1, 1, 5, 11, -1, 1, -11, -5, -1, 1, 6, 19, 14, -15, 14, 19, 6, 1, 1, 7, 29, 47, -19, 19, -47, -29, -7, -1, 1, 8, 41, 104, 37, -56, 37, 104, 41, 8, 1

Offset: 0

Peter Luschny, Jan 18 2015

			Square array starts:
[n\k][0   1   2   3    4     5     6]
[0]   1, -1,  1, -1,   1,   -1,    1, ...
[1]   1,  0, -1,  2,  -3,    4,   -5, ...
[2]   1,  1, -1, -1,   5,  -11,   19, ...
[3]   1,  2,  1, -4,   1,   14,  -47, ...
[4]   1,  3,  5, -1, -15,   19,   37, ...
[5]   1,  4, 11, 14, -19,  -56,  151, ...
[6]   1,  5, 19, 47,  37, -151, -185, ...
The first few rows as a triangle:
1,
1, -1,
1,  0,  1,
1,  1, -1, -1,
1,  2, -1,  2,  1,
1,  3,  1, -1, -3, -1,
1,  4,  5, -4,  5,  4, 1.

Cf. A009940.

T := (n,k) -> k!*coeff(series(exp(-x)*add(binomial(n,j)*x^j, j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k),k=0..6)) od;

T(n,n) = A009940(n).

A330497 a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 0, 1, 26, 1089, 70124, 6495985, 821315214, 136115947009, 28651724077976, 7470040450004001, 2363470644596843330, 892244303052345224641, 396227360441775922668036, 204487588996059177697597969, 121370399839482643287189048374

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Cf. A009940, A025166, A061711, A277423, A307885, A318224, A319392, A330260.

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

A331334 a(n) = n! * [x^n] exp(1 - 1/(2x + 1))/(2x + 1).

Original entry on oeis.org

1, 0, -4, 32, -240, 1792, -11840, 26112, 1589504, -57548800, 1556757504, -39250780160, 973563695104, -24122607992832, 596246557736960, -14477682566889472, 332039052050104320, -6425352382711857152, 53086817854485692416, 4505005802471597015040, -419037805969718712991744

Offset: 0

Peter Luschny, Jan 19 2020

sign

Cf. A331333, A009940.

gf := exp(1 - 1/(2*x + 1))/(2*x + 1): ser := series(gf, x, 32):
seq(n!*coeff(ser, x, n), n=0..20);
# Alternative:
a := proc(n) option remember; if n < 2 then 1 - n else
4*(1 - n)*((n - 1)*a(n - 2) + a(n - 1)) fi end: seq(a(n), n=0..20);

a(n) = 4*(1 - n)*((n - 1)*a(n - 2) + a(n - 1)).
a(n) = (-2)^n*Sum_{k=0..n} A331333(n, k)/(-2)^k.
a(n)/(-2)^n = n!*LaguerreL(n, 1) = A009940(n).

cp's OEIS Frontend

A317365 Expansion of e.g.f. x*exp(x/(1 + x))/(1 + x).

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A331325 a(n) = n!*[x^n] cosh(x/(1-x))/(1-x).

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A331326 a(n) = n!*[x^n] sinh(x/(1 - x))/(1 - x).

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A331688 E.g.f.: exp(-x/(1 - x)) / (1 - 2*x).

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A331725 E.g.f.: exp(x/(1 - x)) / (1 + x).

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A336292 a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(n-k) / (k * ((n-k)!)^2).

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A087860 Expansion of e.g.f.: (1-exp(x/(x-1)))/(1-x).

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A253667 Square array read by ascending antidiagonals, T(n, k) = k![x^k](exp(-x) sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0.

A331334 a(n) = n! * [x^n] exp(1 - 1/(2x + 1))/(2x + 1).