A025168 -id:A025168 - cp's OEIS frontend

A318223 Expansion of e.g.f. exp(x/(1 + 2*x)).

1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151, 236126701, -6169406979, 161388751657, -4327824442967, 120012465557349, -3450029411174219, 102741264191105761, -3160671409312412703, 99982488984008583133, -3230094912866216253971, 105481073534842477321881, -3423260541695907002392679

Offset: 0

Ilya Gutkovskiy, Aug 21 2018

sign

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A002866, A025168, A111884, A122803.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x/(1+2*x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 07 2019

seq(n!*coeff(series(exp(x/(1+2*x)),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 21; CoefficientList[Series[Exp[x/(1 + 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-2)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-2)^(k - 1) k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Join[{1}, Table[(-2)^(n - 1) n! Hypergeometric1F1[1 - n, 2, 1/2], {n, 21}]]

my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1+2*x)))) \\ G. C. Greubel, Feb 07 2019

m = 30; T = taylor(exp(x/(1+2*x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 07 2019

E.g.f.: Product_{k>=1} exp((-2)^(k-1)*x^k).
a(n) = Sum_{k=0..n} (-2)^(n-k)*binomial(n-1,k-1)*n!/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (-2)^(k-1)*k!*binomial(n-1,k-1)*a(n-k).

A341033 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 37, 73, 1, 1, 1, 9, 73, 361, 501, 1, 1, 1, 11, 121, 1009, 4361, 4051, 1, 1, 1, 13, 181, 2161, 17341, 62701, 37633, 1, 1, 1, 15, 253, 3961, 48081, 355951, 1044205, 394353, 1

Offset: 0

Seiichi Manyama, Feb 03 2021

			Square array begins:
  1,   1,    1,     1,     1,      1, ...
  1,   1,    1,     1,     1,      1, ...
  1,   3,    5,     7,     9,     11, ...
  1,  13,   37,    73,   121,    181, ...
  1,  73,  361,  1009,  2161,   3961, ...
  1, 501, 4361, 17341, 48081, 108101, ...

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

Columns 0..7 give A000012, A000262, A025168, A321837, A321847, A321848, A321849, A321850.
Main diagonal gives A293146.
Cf. A253286, A341014.

T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 03 2021 *)

{T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}

{T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}

T(n,k) = Sum_{j=1..n} k^(n-j) * (n!/j!) * binomial(n-1,j-1) for n > 0.

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

A362204 Expansion of e.g.f. exp(x/sqrt(1-2*x)).

Original entry on oeis.org

1, 1, 3, 16, 121, 1176, 13921, 193978, 3106881, 56201176, 1132709041, 25162197006, 610668537073, 16073212005436, 455980333073721, 13868451147012946, 450140785396634881, 15529495879187075088, 567427732311438658081, 21889446540911251445206

Offset: 0

Seiichi Manyama, Apr 11 2023

Cf. A025168, A362163.

Table[n! * Sum[2^(n-k) * Binomial[n-k/2-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 20 2024 *)
Join[{1, 1}, RecurrenceTable[{(-4 + n) (-3 + n) (-2 + n) a[-4 + n] + (-2 + n) (-327 + 290 n - 84 n^2 + 8 n^3) a[-3 + n] + (259 - 299 n + 108 n^2 - 12 n^3) a[-2 + n] + 3 (16 - 13 n + 2 n^2) a[-1 + n] + (5 - n) a[n] == 0, a[2] == 3, a[3] == 16, a[4] == 121, a[5] == 1176}, a, {n, 2, 20}]] (* Vaclav Kotesovec, Feb 20 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/sqrt(1-2*x))))

a(n) = n! * Sum_{k=0..n} (-2)^k * binomial(-(n-k)/2,k)/(n-k)! = n! * Sum_{k=0..n} 2^(n-k) * binomial(n-k/2-1,n-k)/k!.
From Vaclav Kotesovec, Feb 20 2024: (Start)
a(n) ~ 3^(-1/2) * 2^(n - 1/6) * exp(3*2^(-4/3)*n^(1/3) - n) * n^(n - 1/3) * (1 - 3/(16*(n/2)^(1/3))).
Recurrence (for n>5): (n-5)*a(n) = 3*(2*n^2 - 13*n + 16)*a(n-1) - (12*n^3 - 108*n^2 + 299*n - 259)*a(n-2) + (n-2)*(8*n^3 - 84*n^2 + 290*n - 327)*a(n-3) + (n-4)*(n-3)*(n-2)*a(n-4). (End)

A059374 Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

Original entry on oeis.org

1, 3, 2, 13, 18, 6, 73, 156, 108, 24, 501, 1460, 1560, 720, 120, 4051, 15030, 21900, 15600, 5400, 720, 37633, 170142, 315630, 306600, 163800, 45360, 5040, 394353, 2107448, 4763976, 5891760, 4292400, 1834560, 423360, 40320

Offset: 1

Vladeta Jovovic, Jan 28 2001

L'(n, i) are unsigned Lah numbers (Cf. A008297).

			Triangle begins:
  [1],
  [3, 2],
  [13, 18, 6],
  [73, 156, 108, 24],
  [501, 1460, 1560, 720, 120],
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. T(n, 0) = A000262, A025168 (row sums), A000012 (alternating row sums), A059110.

Cf. A008297, A271703.

t[n_, k_] := Sum[ Binomial[n-1, n-i-1]*n!/(n-i)!*Binomial[i, k], {i, 0, n}]; Table[t[n, k], {n, 1, 8}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Mar 22 2013 *)

for(n=1,10, for(k=0,n-1, print1(sum(j=0,n, binomial(j,k)* binomial(n-1,n-j-1)*n!/(n-j)!), ", "))) \\ G. C. Greubel, Jan 29 2018

E.g.f.: exp(x/(1-(1+y)*x))/(1-(1+y)*x)^2. - Vladeta Jovovic, May 10 2003

A079621 Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).

Original entry on oeis.org

1, 4, 1, 24, 12, 1, 192, 144, 24, 1, 1920, 1920, 480, 40, 1, 23040, 28800, 9600, 1200, 60, 1, 322560, 483840, 201600, 33600, 2520, 84, 1, 5160960, 9031680, 4515840, 940800, 94080, 4704, 112, 1, 92897280, 185794560, 108380160, 27095040, 3386880, 225792

Offset: 1

Vladeta Jovovic, Jan 29 2003

Also the Bell transform of A002866(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -1 <= d <= 2). - Shuhei Tsujie, Apr 26 2019

			Triangle begins:
     1;
     4,    1;
    24,   12,   1;
   192,  144,  24,  1;
  1920, 1920, 480, 40, 1;
  ...

P. Bala, The white diamond product of power series.
N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A002866 (first column), A025168 (row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 2^n*(n+1)!, 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2^#*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

E.g.f.: exp(x*y/(1-2*x)).
T(n, k) = n!/k!*binomial(n-1, k-1)*2^(n-k). - Vladeta Jovovic, Sep 24 2003
The n-th row polynomial equals x o (x + 2) o (x + 4) o ... o (x + 2*n), where o is the deformed Hadamard product of power series defined in Bala, section 3.1. - Peter Bala, Jan 18 2018

A331658 E.g.f.: exp(x/(1 - 2*x)) / (1 - x).

Original entry on oeis.org

1, 2, 9, 64, 617, 7446, 107377, 1795844, 34114929, 724822282, 17018900921, 437402060712, 12208463140249, 367629791490014, 11876750557295457, 409663873470828076, 15023747377122799457, 583644746007467984274, 23939828792355240206569, 1033788018952899566018192

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Cf. A000262, A002720, A025167, A025168.

nmax = 19; CoefficientList[Series[Exp[x/(1 - 2 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[Binomial[n, k]^2 k! A000262[n - k], {k, 0, n}]; Table[a[n], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n,k)^2 * k! * A000262(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * k! * A025168(n-k).
a(n) ~ 2^(n + 1/4) * n^(n - 1/4) * exp(-1/4 + sqrt(2*n) - n) * (1 - 23*sqrt(2) / (48*sqrt(n))). - Vaclav Kotesovec, Jan 26 2020

A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, 8, 106, 1954, 46082, 1323064, 44750644, 1741897340, 76672512316, 3764746706176, 203976645319448, 12086590557877144, 777464693554778776, 53948773488864143072, 4016672567726156437744, 319379204127841984947472, 27010128651142535536409360, 2420802590890201251989984128

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049376, A375173, A380257.

Cf. A004211, A025168.

CoefficientList[Series[Exp[ (1/(1-5*x)^(2/5) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-5*x)^(2/5)-1)/2)))

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * (-5)^n * n! * Sum_{k>=0} binomial(-2*k/5,n)/(2^k * k!).

cp's OEIS Frontend

A318223 Expansion of e.g.f. exp(x/(1 + 2*x)).

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A341033 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).

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A362204 Expansion of e.g.f. exp(x/sqrt(1-2*x)).

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A059374 Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

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A079621 Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).

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A331658 E.g.f.: exp(x/(1 - 2*x)) / (1 - x).

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A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

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