A321837 -id:A321837 - cp's OEIS frontend

A321847 E.g.f.: exp(x/(1 - 4*x)).

1, 1, 9, 121, 2161, 48081, 1279801, 39631369, 1398961761, 55422807841, 2434261023721, 117366299630361, 6161301265353169, 349768597919934961, 21347094823271661081, 1393695557886847095721, 96910923898115717350081, 7149718240571434690591809

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2, 3, 4, ... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199
Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000262, A025168, A321837, A321848, A321849, A321850.

seq(coeff(series(factorial(n)*exp(x/(1-4*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[4^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (8n - 7)*a[n - 1] - 16(n - 2)(n - 1) *a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
CoefficientList[Series[Exp[x/(1 - 4*x)], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Dec 07 2018 *)

(a[0] : 1, a[1] : 1, a[n] := (8*n - 7)*a[n-1] - 16*(n-2)*(n-1)*a[n-2], makelist(a[n], n, 0, 20)); /* Franck Maminirina Ramaharo, Nov 27 2018 */

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-4*x)))) \\ Michel Marcus, Nov 25 2018

a(n) = Sum_{k=0..n} 4^(n - k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (8*n - 7)*a(n-1) - 16*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(sqrt(n) - 1/8) * 2^(2*n - 3/2) / (sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321848 E.g.f.: exp(x/(1-5*x)).

Original entry on oeis.org

1, 1, 11, 181, 3961, 108101, 3532651, 134415961, 5834249681, 284391878761, 15378011541451, 913297438474301, 59086483931657161, 4135583008765323181, 311324086814794408811, 25079793551003791168801, 2152597370423901820231201, 196089415332225446044417361

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199
Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000262, A025168, A321837, A321847, A321849, A321850.

seq(coeff(series(factorial(n)*exp(x/(1-5*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[5^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (10 n - 9)*a[n - 1] - 25(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
With[{nn=20},CoefficientList[Series[Exp[x/(1-5x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2024 *)

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-5*x)))) \\ Michel Marcus, Nov 25 2018

a(n) = Sum_{k=0..n} 5^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (10*n-9)*a(n-1) - 25*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(2*sqrt(n/5) - 1/10) * 5^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321849 Expansion of e.g.f.: exp(x/(1-6*x)).

Original entry on oeis.org

1, 1, 13, 253, 6553, 211801, 8201701, 369979093, 19047250993, 1101705494833, 70715424362941, 4987040544656941, 383243311962126793, 31871863566298601353, 2851588139929576342933, 273093945561709965890821, 27871997808801906673665121

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199

Cf. A000262, A025168, A321837, A321847, A321848, A321850.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-6*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 08 2018

[1] cat [&+[6^(n-k)* Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018

seq(coeff(series(factorial(n)*exp(x/(1-6*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[6^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (12n - 11)*a[n - 1] - 36(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-6*x)))) \\ Michel Marcus, Nov 25 2018

f= exp(x/(1-6*x))
g=f.taylor(x,0,13)
L=g.coefficients()
coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
coeffs  # G. C. Greubel, Dec 08 2018

a(n) = Sum_{k=0..n} 6^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (12*n-11)*a(n-1) - 36*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(sqrt(2*n/3) - 1/12) * 6^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321850 E.g.f.: exp(x/(1-7*x)).

Original entry on oeis.org

1, 1, 15, 337, 10081, 376461, 16849351, 878797165, 52324954977, 3501300491641, 260062721279551, 21228108532279881, 1888618754806601665, 181873163575529411077, 18846187172580219099831, 2090754000231731874682021, 247221828043044971020645441

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199

Cf. A000262, A025168, A321837, A321847, A321848, A321849.

[1] cat [&+[7^(n-k)*Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018

seq(coeff(series(factorial(n)*exp(x/(1-7*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[7^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (14n - 13)*a[n - 1] - 49(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-7*x)))) \\ Michel Marcus, Nov 25 2018

a(n) = Sum_{k=0..n} 7^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (14*n-13)*a(n-1) - 49*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(2*sqrt(n/7) - 1/14) * 7^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A362188 Expansion of e.g.f. exp(x/(1-3*x)^(2/3)).

Original entry on oeis.org

1, 1, 5, 43, 513, 7781, 142861, 3075255, 75879553, 2110145833, 65275127541, 2222656864451, 82595058938305, 3325666654250253, 144214230714973693, 6700048267934377231, 331988586859256529921, 17475202293073669341905, 973765103770578798536293

Offset: 0

Seiichi Manyama, Apr 11 2023

Cf. A321837, A362164, A362205.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-3*x)^(2/3))))

a(n) = n! * Sum_{k=0..n} (-3)^k * binomial(-2*(n-k)/3,k)/(n-k)! = n! * Sum_{k=0..n} 3^(n-k) * binomial(n-k/3-1,n-k)/k!.

A362205 Expansion of e.g.f. exp(x/(1-3*x)^(1/3)).

Original entry on oeis.org

1, 1, 3, 19, 185, 2401, 38731, 745123, 16630769, 422157025, 12005107091, 377957000851, 13048046175913, 490052749100929, 19890724260375515, 867582126490694371, 40467070835396193761, 2009901604798183428673, 105901641663222888913699

Offset: 0

Seiichi Manyama, Apr 11 2023

Cf. A321837, A362164, A362188.

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

a(n) = n! * Sum_{k=0..n} (-3)^k * binomial(-(n-k)/3,k)/(n-k)! = n! * Sum_{k=0..n} 3^(n-k) * binomial(n-2*k/3-1,n-k)/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.

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A321847 E.g.f.: exp(x/(1 - 4*x)).

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A321848 E.g.f.: exp(x/(1-5*x)).

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A321849 Expansion of e.g.f.: exp(x/(1-6*x)).

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A321850 E.g.f.: exp(x/(1-7*x)).

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A362188 Expansion of e.g.f. exp(x/(1-3*x)^(2/3)).

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A362205 Expansion of e.g.f. exp(x/(1-3*x)^(1/3)).

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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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