A027710 -id:A027710 - cp's OEIS frontend

A276506 E.g.f.: exp(9*(exp(x)-1)).

1, 9, 90, 981, 11511, 144108, 1911771, 26730981, 392209380, 6016681467, 96202473183, 1599000785730, 27563715220509, 491777630207037, 9064781481234546, 172346601006842337, 3375007346801025099, 67983454804021156548, 1406921223577401454239, 29881379179971835132761

Offset: 0

Vincenzo Librandi, Sep 17 2016

nonn

Number of ways of placing n labeled balls into n unlabeled (but 9-colored) boxes.

Alois P. Heinz, Table of n, a(n) for n = 0..514

Cf. similar sequences with e.g.f. exp(k*(exp(x)-1)): A001861 (k=2), A027710 (k=3), A078944 (k=4), A144180 (k=5) A144223 (k=6), A144263 (k=7), A221159 (k=8), this sequence (k=9), A276507 (k=10).

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*9)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 25 2017

Mathematica
```
Table[BellB[n, 9], {n, 0, 30}]
```

my(x='x+O('x^99)); Vec(serlaplace(exp(9*(exp(x)-1)))) \\ Altug Alkan, Sep 17 2016

G.f.: A(x) satisfies 9*(x/(1-x))*A(x/(1-x)) = A(x)-1; nine times the binomial transform equals this sequence shifted one place left.

A335981 Expansion of e.g.f. exp(3 * (1 - exp(-x)) + x).

Original entry on oeis.org

1, 4, 13, 31, 40, -23, -95, 490, 823, -8393, 3766, 174775, -658787, -2751404, 34033297, -55552037, -1170734432, 9362348365, 3277050925, -562286419646, 3848880970147, 8815342530739, -356804325202730, 2389771436686339, 8677476137729929, -302470260552857660

Offset: 0

Ilya Gutkovskiy, Jul 03 2020

sign

Cf. A027710, A078940, A109747, A309084, A335868, A335980, A335982.

nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 3 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

a(n) = exp(3) * (-1)^n * Sum_{k>=0} (-3)^k * (k - 1)^n / k!.

a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).

A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k) a(n-2*k-1).

Original entry on oeis.org

1, 3, 9, 30, 117, 516, 2493, 13152, 75177, 460272, 3003921, 20806176, 152114013, 1169842368, 9435180357, 79553524224, 699531782481, 6400932102912, 60820145019801, 599036357936640, 6105903392066373, 64309189153428480, 698936466350352717, 7828833281592926208, 90270159223293364473

Offset: 0

Ilya Gutkovskiy, Mar 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..565

Cf. A027710, A080527, A107403.

Cf. A003724, A136630, A352279.

a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Exp[3 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

E.g.f.: exp( 3 * sinh(x) ).

a(n) = Sum_{k=0..n} 3^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 9, 27, 144, 945, 6273, 44226, 339399, 2796795, 24387786, 223853355, 2159078445, 21827316888, 230536050165, 2536213188519, 28994911890048, 343806474384045, 4220933769308205, 53566838971016418, 701650841036287275, 9473067208871584407

Offset: 0

Seiichi Manyama, Oct 08 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..547
Eric Weisstein's World of Mathematics, Bell Polynomial.

Column k=9 of A357681.
Cf. A024430, A065143, A264036, A357615.
Cf. A027710, A357649, A357668.

my(x='x+O('x^30)); Vec(serlaplace(cosh(3*(exp(x)-1))))

a(n) = sum(k=0, n\2, 9^k*stirling(n, 2*k, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 3)+Bell_poly(n, -3)))/2;

E.g.f.: cosh( 3 * (exp(x) - 1) ).
a(n) = Sum_{k=0..floor(n/2)} 9^k * Stirling2(n,2*k).
a(n) = ( Bell_n(3) + Bell_n(-3) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 9 * Sum_{k=0..n-1} binomial(n-1, k) * A357668(k).

A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).

Original entry on oeis.org

1, 5, 28, 173, 1165, 8468, 65923, 546197, 4791214, 44301143, 430158397, 4372004546, 46381674085, 512328076385, 5879362011436, 69958289731457, 861605015493073, 10965899141265500, 144018319806024991, 1949190279770578145, 27153595018237222774

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000110, A005493, A035009, A078940, A355247, A367888.

nmax = 20; CoefficientList[Series[Exp[3 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 2^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 + x * ( 2 * A(x) + 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k+2)^n / k!.
a(0) = 1; a(n) = 2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A027710(k).

A357668 Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.

Original entry on oeis.org

0, 1, 1, 10, 55, 307, 2026, 14779, 114157, 933616, 8110261, 74525167, 719925328, 7279859485, 76855303981, 845280487018, 9663800287483, 114601481983855, 1407040763488354, 17856103120048783, 233883061849700137, 3157648445216335528, 43887908697233605489

Offset: 0

Seiichi Manyama, Oct 08 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..547
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A024429, A264037, A357572, A357598.

Cf. A027710, A356572, A357667.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(3*(exp(x)-1))/3)))

a(n) = sum(k=0, (n-1)\2, 9^k*stirling(n, 2*k+1, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 3)-Bell_poly(n, -3)))/6;

a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(3) - Bell_n(-3) )/6, where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357667(k).

A367890 Expansion of e.g.f. exp(3*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 3, 3, 30, 93, 633, 3342, 22809, 156063, 1183872, 9453711, 80455125, 721576560, 6809391111, 67332650007, 695777512638, 7493572404345, 83926492573341, 975467527353750, 11744536832206149, 146234590864310019, 1880198749437144456, 24928860500681953683

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A027710, A194689, A346738, A355253, A355254, A367888, A367891.

nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - 3 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-3)^n / k!.
a(0) = 1; a(n) = 3 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A027710(k).

A375871 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)) - 1) ).

Original entry on oeis.org

1, 3, 30, 543, 14493, 515001, 22930869, 1229340027, 77151412902, 5551075890453, 450607640485269, 40745592546015495, 4061982705195354033, 442649982865922396337, 52351468801767526253538, 6678605910447082873015923, 914198409310749883430655441

Offset: 0

Seiichi Manyama, Sep 01 2024

nonn

Cf. A349683, A375870, A375872.

Cf. A027710, A370057.

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

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A349683.

a(n) = 3 * Sum_{k=0..n} (3*n+3)^(k-1) * Stirling2(n,k).

A375869 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)^(2/3)) - 1) ).

Original entry on oeis.org

1, 3, 24, 327, 6405, 164856, 5276523, 202365351, 9055962270, 463552982301, 26725378964169, 1714193590625478, 121100759112660789, 9344673700445352639, 782093803535217656256, 70570503124491323693523, 6829746633650550406177713, 705729134240394228512985960

Offset: 0

Seiichi Manyama, Sep 01 2024

nonn

Cf. A349598, A375868.

Cf. A027710, A370055.

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

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A349598.

a(n) = 3 * Sum_{k=0..n} (2*n+3)^(k-1) * Stirling2(n,k).

cp's OEIS Frontend

A276506 E.g.f.: exp(9*(exp(x)-1)).

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A335981 Expansion of e.g.f. exp(3 * (1 - exp(-x)) + x).

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A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

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A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).

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A367889 Expansion of e.g.f. exp(3*(exp(x) - 1) + 2*x).

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A357668 Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.

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A367890 Expansion of e.g.f. exp(3*(exp(x) - 1 - x)).

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A375871 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)) - 1) ).

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A375869 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)^(2/3)) - 1) ).

A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k) a(n-2*k-1).

A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).