A094418 -id:A094418 - cp's OEIS frontend

A238464 Generalized ordered Bell numbers Bo(7,n).

1, 7, 105, 2359, 70665, 2646007, 118893705, 6232661239, 373405001865, 25167452766967, 1884759251911305, 155262005162499319, 13952854271421949065, 1358385484966283220727, 142418920493123648992905, 15998363870912950298468599

Offset: 0

Vincenzo Librandi, Mar 17 2014

nonn

Row 7 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.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(8 - 7*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Bruno Berselli, Mar 17 2014

t=30; Range[0, t]! CoefficientList[Series[1/(8 - 7 Exp[x]), {x, 0, t}], x]

x='x+O('x^66); Vec(serlaplace(1/(8 - 7*exp(x)))) \\ Joerg Arndt, Mar 17 2014

E.g.f.: 1/(8 - 7*exp(x)).
a(n) ~ n! / (8*(log(8/7))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 7*a(n-1) - 8*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A344840 a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 5, 35, 265, 2195, 19625, 187755, 1909185, 20521515, 232124745, 2752591475, 34108980105, 440444019835, 5912197332865, 82320781521195, 1186703083508025, 17680850448587155, 271845880552898985, 4307188044378111915, 70236616096770062945, 1177406236243423738475

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A040027, A094418, A144180, A343523, A343975, A344735, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 5 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1 + 5 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 5 * x * A(x/(1 - x)) / (1 - x)^2.

A365588 Expansion of e.g.f. 1 / (1 + 5 * log(1-x)).

Original entry on oeis.org

1, 5, 55, 910, 20080, 553870, 18333050, 707959800, 31244562600, 1551289408800, 85579293493200, 5193226343508000, 343790892166398000, 24655487205067386000, 1904221630155352038000, 157574022827034258192000, 13908505761692419540320000

Offset: 0

Seiichi Manyama, Sep 10 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..350

Column k=5 of A320079.
Cf. A346987, A365585, A365586, A365587.
Cf. A094418.

a[n_] := Sum[5^k * k! * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)

a(n) = sum(k=0, n, 5^k*k!*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} 5^k * k! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 5 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * exp(4*n/5) * (exp(1/5) - 1)^(n+1)). - Vaclav Kotesovec, Nov 11 2023

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).

A365568 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(2/5).

Original entry on oeis.org

1, 2, 16, 212, 3964, 95804, 2840140, 99760124, 4050900268, 186700658972, 9628444876108, 549349531209404, 34355463031007596, 2336935606239856988, 171779270567736231052, 13568895740353218626300, 1146225546710339427328684, 103113032296428007394503580

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094418, A346984, A365569, A365570.

a[n_] := Sum[Product[5*j + 2, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)

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

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+2)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (5 - 3*k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(Pi) * 2^(1/10) * n^(n - 1/10) / (3^(2/5) * Gamma(2/5) * exp(n) * log(6/5)^(n + 2/5)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 2*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A365569 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(3/5).

Original entry on oeis.org

1, 3, 27, 387, 7659, 193491, 5948091, 215446563, 8984708235, 423944899443, 22328393101659, 1298429924941251, 82625791930962219, 5711012035686681363, 426058604580805219323, 34121803137713388036963, 2919847869159667841599947, 265868538017899566748612275

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094418, A346984, A365568, A365570.

a[n_] := Sum[Product[5*j + 3, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
With[{nn=20},CoefficientList[Series[1/(6-5*Exp[x])^(3/5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 03 2024 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+3)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+3)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (5 - 2*k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 1/10) / (6^(3/5) * Gamma(3/5) * exp(n) * log(6/5)^(n + 3/5)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 3*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A365570 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(4/5).

Original entry on oeis.org

1, 4, 40, 616, 12856, 338728, 10781176, 402250216, 17213590840, 831013114792, 44675458306168, 2646758624166760, 171319908334752184, 12028779733435667752, 910538645035885918456, 73918475291961325824232, 6406179168820339231897144

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094418, A346984, A365568, A365569.

a[n_] := Sum[Product[5*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, 5*j+4)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+4)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (5 - k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/10) / (6^(4/5) * Gamma(4/5) * exp(n) * log(6/5)^(n + 4/5)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 4*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

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

A355113 Expansion of e.g.f. 5 / (6 - 5x - exp(5x)).

Original entry on oeis.org

1, 2, 13, 133, 1779, 29565, 589705, 13728695, 365295695, 10934634985, 363678872325, 13305294463275, 531030788556475, 22960273845453725, 1069101897816615425, 53336480697298243375, 2838300249311563302375, 160480124820425410172625, 9607441647405962075600125

Offset: 0

Ilya Gutkovskiy, Jun 19 2022

nonn

Cf. A003577, A006155, A094418, A355110, A355111, A355112, A355114.

nmax = 18; CoefficientList[Series[5/(6 - 5 x - Exp[5 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 5^(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) * 5^(k-1) * a(n-k).

a(n) ~ n! / ((1 + LambertW(exp(6))) * ((6 - LambertW(exp(6)))/5)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A365604 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x)).

Original entry on oeis.org

1, 5, 45, 610, 11020, 248870, 6744350, 213233400, 7704814200, 313199930400, 14146162064400, 702826758144000, 38093116667766000, 2236695336601458000, 141433354184701746000, 9582086196220281456000, 692463727252196674560000

Offset: 0

Seiichi Manyama, Sep 11 2023

nonn

Column k=5 of A320080.
Cf. A347022, A365601, A365602, A365603.
Cf. A094418, A365588.

a[n_] := Sum[5^k * k! * StirlingS1[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)
With[{nn=20},CoefficientList[Series[1/(1-5*Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 05 2025 *)

a(n) = sum(k=0, n, 5^k*k!*stirling(n, k, 1));

a(n) = Sum_{k=0..n} 5^k * k! * Stirling1(n,k).

a(0) = 1; a(n) = 5 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

cp's OEIS Frontend

A238464 Generalized ordered Bell numbers Bo(7,n).

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A344840 a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A365588 Expansion of e.g.f. 1 / (1 + 5 * log(1-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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A365568 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(2/5).

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A365569 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(3/5).

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A365570 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(4/5).

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

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Magma

A355113 Expansion of e.g.f. 5 / (6 - 5x - exp(5x)).