A094419 -id:A094419 - cp's OEIS frontend

A384521 Expansion of e.g.f. 5/(7 - 2exp(5x)).

1, 2, 18, 218, 3474, 69290, 1659330, 46359770, 1480241970, 53171142410, 2122154748450, 93168872862650, 4462242691496850, 231524863130863850, 12936797161953970050, 774495903492069700250, 49458416187322116299250, 3355754824852804221058250, 241081466990843266748993250

Offset: 0

Seiichi Manyama, Jun 01 2025

nonn

Wikipedia, Polylogarithm.

Cf. A094419, A384514, A384522, A384523, A384524.

PARI
```
a(n) = (-5)^(n+1)*polylog(-n, 7/2)/7;
```

a(n) = (-5)^(n+1)/7 * Li_{-n}(7/2), where Li_{n}(x) is the polylogarithm function.
a(n) = 5^(n+1)/7 * Sum_{k>=0} k^n * (2/7)^k.
a(n) = Sum_{k=0..n} 2^k * 5^(n-k) * k! * Stirling2(n,k).
a(n) = (2/7) * Sum_{k=0..n} 7^k * (-5)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 2 * Sum_{k=1..n} 5^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2 * a(n-1) + 7 * Sum_{k=1..n-1} (-5)^(k-1) * binomial(n-1,k) * a(n-k).

A384522 Expansion of e.g.f. 4/(7 - 3exp(4x)).

Original entry on oeis.org

1, 3, 30, 426, 8040, 189768, 5375280, 177632976, 6708685440, 285038686848, 13456362881280, 698786099602176, 39586707755811840, 2429498408440009728, 160571526535426529280, 11370607719608891467776, 858870213271187908362240, 68928740686010010238353408

Offset: 0

Seiichi Manyama, Jun 01 2025

nonn

Wikipedia, Polylogarithm.

Cf. A094419, A384514, A384521, A384523, A384524.

With[{nn=20},CoefficientList[Series[4/(7-3Exp[4x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 21 2025 *)

PARI
```
a(n) = (-4)^(n+1)*polylog(-n, 7/3)/7;
```

a(n) = (-4)^(n+1)/7 * Li_{-n}(7/3), where Li_{n}(x) is the polylogarithm function.
a(n) = 4^(n+1)/7 * Sum_{k>=0} k^n * (3/7)^k.
a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * k! * Stirling2(n,k).
a(n) = (3/7) * Sum_{k=0..n} 7^k * (-4)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} 4^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3 * a(n-1) + 7 * Sum_{k=1..n-1} (-4)^(k-1) * binomial(n-1,k) * a(n-k).

A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 10, 3, 1, 0, 75, 74, 21, 4, 1, 0, 541, 730, 219, 36, 5, 1, 0, 4683, 9002, 3045, 484, 55, 6, 1, 0, 47293, 133210, 52923, 8676, 905, 78, 7, 1, 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1, 0, 7087261, 45375130, 26857659, 5227236, 544505, 39390, 2359, 136, 9, 1

Offset: 0

Peter Luschny, May 21 2021

The array rows are recursively generated by applying the Akiyama-Tanigawa algorithm to the powers (see the Python implementation below). In this way the array becomes the image of A004248 under the AT-transformation when applied to the columns of A004248. This makes the array closely linked to A371761, which is generated in the same way, but applied to the rows of A004248. - Peter Luschny, Apr 27 2024

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,      1;
[3] 0, 3,      2,       1;
[4] 0, 13,     10,      3,       1;
[5] 0, 75,     74,      21,      4,      1;
[6] 0, 541,    730,     219,     36,     5,     1;
[7] 0, 4683,   9002,    3045,    484,    55,    6,    1;
[8] 0, 47293,  133210,  52923,   8676,   905,   78,   7,   1;
[9] 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1;
.
Seen as an array A(n, k) = T(n + k, n):
[0] [1, 0,   0,    0,     0,       0,         0, ...  A000007
[1] [1, 1,   3,   13,    75,     541,      4683, ...  A000670
[2] [1, 2,  10,   74,   730,    9002,    133210, ...  A004123
[3] [1, 3,  21,  219,  3045,   52923,   1103781, ...  A032033
[4] [1, 4,  36,  484,  8676,  194404,   5227236, ...  A094417
[5] [1, 5,  55,  905, 19855,  544505,  17919055, ...  A094418
[6] [1, 6,  78, 1518, 39390, 1277646,  49729758, ...  A094419
[7] [1, 7, 105, 2359, 70665, 2646007, 118893705, ...  A238464

Variant of the array is A094416 (which has column 0 and row 0 missing).
The coefficients of the Fubini polynomials are A131689.
Cf. A094420 (main diagonal of array), A372346 (row sums), A004248, A371761.

F := proc(n) option remember; if n = 0 then return 1 fi:
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
seq(seq(subs(x = k, F(n - k)), k = 0..n), n = 0..10);

F[n_] := F[n] = If[n == 0, 1,
   Expand[Sum[Binomial[n, k]*F[n - k]*x, {k, 1, n}]]];
Table[Table[F[n - k] /. x -> k, {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 06 2024, after Peter Luschny *)

# Computes the triangle.
@cached_function
def F(n):
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
def Fval(n): return [F(n - k).substitute(x = k) for k in (0..n)]
for n in range(10): print(Fval(n))

# Computes the square array using the Akiyama-Tanigawa algorithm.
def ATFubini(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = (n + 1)**k  # Chancing this to R[k] = k**n generates A371761.
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(8): print([n], ATFubini(n, 7))  # Peter Luschny, Apr 27 2024

T(n, k) = (n - k)! * [x^(n - k)] (1 / (1 + k * (1 - exp(x)))).

T(2*n, n) = A094420(n).

A365556 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(2/3).

Original entry on oeis.org

1, 4, 44, 764, 18204, 551644, 20291804, 877970524, 43680345564, 2456429581404, 154072160204764, 10663000409493084, 807124301044917724, 66329628496719183964, 5881222650127663682524, 559616682597652939940444, 56879286407092006924382684

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094419, A346985, A354252, A365555, A365557.

a[n_] := Sum[Product[6*j + 4, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+4)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+2)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+4)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 2*k/n) * binomial(n,k) * a(n-k).
O.g.f. (conjectural): 1/(1 - 4*x/(1 - 7*x/(1 - 10*x/(1 - 14*x/(1 - 16*x/(1 - 21*x/(1 - ... - (6*n - 2)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Sep 24 2023
a(n) ~ Gamma(1/3) * sqrt(3) * n^(n + 1/6) / (sqrt(2*Pi) * 7^(2/3) * exp(n) * log(7/6)^(n + 2/3)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 4*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A365557 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(5/6).

Original entry on oeis.org

1, 5, 60, 1105, 27505, 862900, 32665935, 1448431605, 73618245530, 4219213176975, 269178309769385, 18919087590749230, 1452439246800583805, 120926788470961893425, 10852505784073190637460, 1044349665968997385498605, 107273533723839304683589205

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094419, A346985, A354252, A365555, A365556.

a[n_] := Sum[Product[6*j + 5, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+5)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+5)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - k/n) * binomial(n,k) * a(n-k).
O.g.f. (conjectural): 1/(1 - 5*x/(1 - 7*x/(1 - 11*x/(1 - 14*x/(1 - 17*x/(1 - 21*x/(1 - ... - (6*n - 1)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Sep 24 2023
a(n) ~ Gamma(1/3)^2 * sqrt(3) * n^(n + 1/3) / (14^(5/6) * Pi * exp(n) * log(7/6)^(n + 5/6)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 5*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A384523 Expansion of e.g.f. 3/(7 - 4exp(3x)).

Original entry on oeis.org

1, 4, 44, 708, 15180, 406884, 13087404, 491114628, 21062220300, 1016197112484, 54476506976364, 3212426755972548, 206654933095516620, 14401921040252826084, 1080885666078491553324, 86916516692600836638468, 7455102038197447378720140, 679412933203279242481083684

Offset: 0

Seiichi Manyama, Jun 01 2025

nonn

Wikipedia, Polylogarithm.

Cf. A094419, A384514, A384521, A384522, A384524.

PARI
```
a(n) = (-3)^(n+1)*polylog(-n, 7/4)/7;
```

a(n) = (-3)^(n+1)/7 * Li_{-n}(7/4), where Li_{n}(x) is the polylogarithm function.
a(n) = 3^(n+1)/7 * Sum_{k>=0} k^n * (4/7)^k.
a(n) = Sum_{k=0..n} 4^k * 3^(n-k) * k! * Stirling2(n,k).
a(n) = (4/7) * Sum_{k=0..n} 7^k * (-3)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 4 * Sum_{k=1..n} 3^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4 * a(n-1) + 7 * Sum_{k=1..n-1} (-3)^(k-1) * binomial(n-1,k) * a(n-k).

A384524 Expansion of e.g.f. 2/(7 - 5exp(2x)).

Original entry on oeis.org

1, 5, 60, 1070, 25440, 756080, 26964960, 1121963120, 53351831040, 2854122433280, 169649803023360, 11092432778385920, 791204615734640640, 61138238969353748480, 5087702653663698677760, 453621615686933964830720, 43141424825262182799114240, 4359374368561019960377671680

Offset: 0

Seiichi Manyama, Jun 01 2025

nonn

Wikipedia, Polylogarithm.

Cf. A094419, A384514, A384521, A384522, A384523.

A384524:=proc(n)
    add(5^k * 2^(n-k) * k! * combinat[stirling2](n,k) ,k=0..n) ;
end proc:
seq(A384524(n), n=0..40); # R. J. Mathar, Jun 04 2025

PARI
```
a(n) = (-2)^(n+1)*polylog(-n, 7/5)/7;
```

a(n) = (-2)^(n+1)/7 * Li_{-n}(7/5), where Li_{n}(x) is the polylogarithm function.
a(n) = 2^(n+1)/7 * Sum_{k>=0} k^n * (5/7)^k.
a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * k! * Stirling2(n,k).
a(n) = (5/7) * Sum_{k=0..n} 7^k * (-2)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 5 * Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5 * a(n-1) + 7 * Sum_{k=1..n-1} (-2)^(k-1) * binomial(n-1,k) * a(n-k).

A238465 Generalized ordered Bell numbers Bo(8,n).

Original entry on oeis.org

1, 8, 136, 3464, 117640, 4993928, 254396296, 15119104904, 1026912225160, 78468091562888, 6662087721342856, 622186077361470344, 63389713864392140680, 6996476832548305415048, 831619554631233264449416, 105909083171031626820475784

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 8 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(9 - 8*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t = 30; Range[0, t]! CoefficientList[Series[1/(9 - 8 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(9 - 8*exp(x)).
a(n) ~ n! / (9*(log(9/8))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 8*a(n-1) - 9*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A355114 Expansion of e.g.f. 6 / (7 - 6x - exp(6x)).

Original entry on oeis.org

1, 2, 14, 156, 2256, 40416, 869040, 21817440, 626063616, 20210176512, 724888631808, 28599923045376, 1230970377166848, 57397448756994048, 2882187551571941376, 155065468075097960448, 8898907099302329647104, 542609247778976191610880, 35031706496702707368591360

Offset: 0

Ilya Gutkovskiy, Jun 19 2022

nonn

Cf. A003578, A006155, A094419, A355110, A355111, A355112, A355113.

nmax = 18; CoefficientList[Series[6/(7 - 6 x - Exp[6 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 6^(k-1) * a(n-k).

a(n) ~ n! / ((1 + LambertW(exp(7))) * ((7 - LambertW(exp(7)))/6)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A365555 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/3).

Original entry on oeis.org

1, 2, 18, 274, 5938, 167122, 5786418, 237857874, 11319677618, 612109819602, 37069480301618, 2485356833141074, 182753029186750898, 14623552941626800082, 1265002802597606144818, 117633823750542653153874, 11701922865351577653913778

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094419, A346985, A354252, A365556, A365557.

a[n_] := Sum[Product[6*j + 2, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)
With[{nn=20},CoefficientList[Series[1/(7-6Exp[x])^(1/3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 21 2024 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+2)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+2)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 4*k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n - 1/6) / (Gamma(1/3) * 7^(1/3) * exp(n) * log(7/6)^(n + 1/3)). - Vaclav Kotesovec, Sep 09 2023
From Peter Bala, Sep 24 2023: (Start)
O.g.f. (conjectural): 1/(1 - 2*x/(1 - 7*x/(1 - 8*x/(1 - 14*x/(1 - 14*x/(1 - 21*x/(1 - ... - (6*n - 4)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction).
More generally, it appears that the o.g.f. of the sequence whose e.g.f. is 1/(r+1 - r*exp(s*x))^(m/s) corresponds to the S-fraction 1/(1 - r*m*x/(1 - s*(r+1)*x/(1 - r*(m+s)*x/(1 - 2*s(r+1)*x/(1 - r*(m+2*s)*x/(1 - 3*s(r+1)*x/( 1 - ... ))))))). This is the case r = 6, s = 1, m = 1/3. (End)
a(0) = 1; a(n) = 2*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

cp's OEIS Frontend

A384521 Expansion of e.g.f. 5/(7 - 2*exp(5*x)).

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A384522 Expansion of e.g.f. 4/(7 - 3*exp(4*x)).

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A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

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Maple

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A365556 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(2/3).

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Mathematica

PARI

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A365557 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(5/6).

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Mathematica

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A384523 Expansion of e.g.f. 3/(7 - 4*exp(3*x)).

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A384524 Expansion of e.g.f. 2/(7 - 5*exp(2*x)).

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Maple

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A238465 Generalized ordered Bell numbers Bo(8,n).

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Magma

Mathematica

A384521 Expansion of e.g.f. 5/(7 - 2exp(5x)).

A384522 Expansion of e.g.f. 4/(7 - 3exp(4x)).

A384523 Expansion of e.g.f. 3/(7 - 4exp(3x)).

A384524 Expansion of e.g.f. 2/(7 - 5exp(2x)).

A355114 Expansion of e.g.f. 6 / (7 - 6x - exp(6x)).