A032033 -id:A032033 - cp's OEIS frontend

A367473 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^3.

1, 9, 117, 1953, 39645, 946089, 25926597, 801869553, 27618402285, 1048096422009, 43444114011477, 1952712851250753, 94592798546953725, 4912513525545837129, 272265236648295312357, 16039329591716508497553, 1000809252891040145821965

Offset: 0

Seiichi Manyama, Nov 19 2023

nonn

Cf. A032033, A367472.

Cf. A226515, A367471.

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

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * Stirling2(n,k).
a(0) = 1; a(n) = 3*Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 9*a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

A375949 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(4/3).

Original entry on oeis.org

1, 4, 32, 368, 5520, 102064, 2242832, 57095728, 1652211600, 53559908784, 1922581295632, 75700072208688, 3243905700776080, 150289130386531504, 7485459789379535632, 398857142195958963248, 22639650637589839298960, 1363772478150606703714224

Offset: 0

Seiichi Manyama, Sep 03 2024

nonn

Cf. A032033, A346982, A365558, A375952.

Cf. A005649, A007559, A375948.

nmax=17; CoefficientList[Series[1 / (4 - 3 * Exp[x])^(4/3),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)

a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k+1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} A007559(k+1) * Stirling2(n,k).

a(n) ~ 3 * sqrt(Pi) * n^(n + 5/6) / (2^(13/6) * Gamma(1/3) * log(4/3)^(n + 4/3) * exp(n)). - Vaclav Kotesovec, Sep 06 2024

A384325 Expansion of Product_{k>=1} 1/(1 - k*x)^((3/4)^k).

Original entry on oeis.org

1, 12, 114, 1084, 11319, 136920, 1981228, 34705656, 731268315, 18203860748, 524073230394, 17111173850652, 623571696107069, 25046605210733184, 1097919954149781264, 52109508350206511840, 2660615337817983390318, 145353541761618312219336

Offset: 0

Seiichi Manyama, May 26 2025

nonn

Cf. A084785, A384305, A384324, A384326.

Cf. A032033, A090353.

terms = 20; A[] = 1; Do[A[x] = -3*A[x] + 4*A[x/(1-x)]^(3/4) / (1-x)^3 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 27 2025 *)

my(N=20, x='x+O('x^N)); Vec(exp(4*sum(k=1, N, sum(j=0, k, 3^j*j!*stirling(k, j, 2))*x^k/k)))

G.f. A(x) satisfies A(x) = A(x/(1-x))^(3/4) / (1-x)^3.
G.f.: exp(4 * Sum_{k>=1} A032033(k) * x^k/k).
G.f.: B(x)^12, where B(x) is the g.f. of A090353.
a(n) ~ (n-1)! / log(4/3)^(n+1). - Vaclav Kotesovec, May 27 2025

A375952 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(5/3).

Original entry on oeis.org

1, 5, 45, 565, 9085, 177925, 4106445, 109105365, 3279219485, 109983317925, 4071784884845, 164919693538165, 7253726995805885, 344284133391481925, 17538600019076063245, 954467594134586386965, 55263075631036363208285, 3391909484128563111709925

Offset: 0

Seiichi Manyama, Sep 03 2024

nonn

Cf. A032033, A346982, A365558, A375949.

Cf. A008544.

nmax=17; CoefficientList[Series[1 / (4 - 3 * Exp[x])^(5/3),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)

a008544(n) = prod(k=0, n-1, 3*k+2);
a(n) = sum(k=0, n, a008544(k+1)*stirling(n, k, 2))/2;

a(n) = (1/2) * Sum_{k=0..n} A008544(k+1) * Stirling2(n,k).

A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 3^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 21, 9, 1, 1, 219, 657, 27, 1, 1, 3045, 119241, 22869, 81, 1, 1, 52923, 40365873, 80850987, 836001, 243, 1, 1, 1103781, 21955523049, 747786838869, 60579666801, 31436181, 729, 1, 1, 26857659, 17512689629457, 14298291269335467, 16117269494868801, 48066954848379, 1204022961, 2187, 1

Offset: 0

Seiichi Manyama, May 27 2025

			Square array begins:
  1,  1,      1,           1,                 1, ...
  1,  3,     21,         219,              3045, ...
  1,  9,    657,      119241,          40365873, ...
  1, 27,  22869,    80850987,      747786838869, ...
  1, 81, 836001, 60579666801, 16117269494868801, ...

Columns k=0..2 give A000012, A000244, 3^n * A084768(n).
Rows n=0..1 give A000012, A032033.
Cf. A262809, A384362.

a(n, k) = sum(i=0, k*n, 3^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));

A(n,k) = (1/4) * Sum_{j>=0} (3/4)^j * binomial(j,n)^k.

A238466 Generalized ordered Bell numbers Bo(9,n).

Original entry on oeis.org

1, 9, 171, 4869, 184851, 8772309, 499559571, 33190014069, 2520110222451, 215270320769109, 20431783142389971, 2133148392099721269, 242954208655633344051, 29977118969127060357909, 3983272698956314883956371, 567091857051921058649396469

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 9 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/(10 - 9*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(10 - 9 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(10 - 9*exp(x)).
a(n) ~ n! / (10*(log(10/9))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 9*a(n-1) - 10*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A238467 Generalized ordered Bell numbers Bo(10,n).

Original entry on oeis.org

1, 10, 210, 6610, 277410, 14553010, 916146210, 67285818610, 5647734061410, 533307215001010, 55954905981282210, 6457903731351210610, 813080459351919805410, 110901542660769629769010, 16290196917457939734258210

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 10 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/(11 - 10*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(11 - 10 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(11 - 10*exp(x)).
a(n) ~ n! / (11*(log(11/10))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 10*a(n-1) - 11*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A368322 Expansion of e.g.f. exp(2x) / (4 - 3exp(x)).

Original entry on oeis.org

1, 5, 37, 389, 5413, 94085, 1962277, 47746949, 1327769893, 41538664325, 1443908686117, 55210237509509, 2302968844974373, 104068337416767365, 5064468256286449957, 264065894676248072069, 14686540175450593986853, 867871886679723760867205

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A032033, A201354, A368323, A368324.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=2, t=3) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

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

a(n) = (16/9)*A032033(n) - (1/3)*(1 + (4/3)*0^n).

A368323 Expansion of e.g.f. exp(3x) / (4 - 3exp(x)).

Original entry on oeis.org

1, 6, 48, 516, 7212, 125436, 2616348, 63662556, 1770359772, 55384885596, 1925211581148, 73613650011996, 3070625126631132, 138757783222353756, 6752624341715261148, 352087859568330751836, 19582053567267458627292, 1157162515572965014445916

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A032033, A201354, A368322, A368324.

With[{nn=20},CoefficientList[Series[Exp[3x]/(4-3Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 18 2025 *)

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=3, t=3) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

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

a(n) = (64/27)*A032033(n) - (1/3)*(2^n + 4/3 + (16/9)*0^n).

A368324 Expansion of e.g.f. exp(4x) / (4 - 3exp(x)).

Original entry on oeis.org

1, 7, 61, 679, 9589, 167167, 3488221, 84882679, 2360477509, 73846507567, 2566948755181, 98151533290279, 4094166835331029, 185010377629273567, 9003499122285420541, 469450479424436219479, 26109404756356597154149, 1542883354097286642881167

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A032033, A201354, A368322, A368323.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=4, t=3) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

a(n) = 4^n + 3 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (256/81)*A032033(n) - (1/3)*(3^n + (4/3)*2^n + 16/9 + (64/27)*0^n).

cp's OEIS Frontend

A367473 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^3.

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A375949 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(4/3).

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A384325 Expansion of Product_{k>=1} 1/(1 - k*x)^((3/4)^k).

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A375952 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(5/3).

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A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..k*n} 3^i * Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

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A238466 Generalized ordered Bell numbers Bo(9,n).

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Magma

Mathematica

Formula

A238467 Generalized ordered Bell numbers Bo(10,n).

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Magma

Mathematica

Formula

A368322 Expansion of e.g.f. exp(2*x) / (4 - 3*exp(x)).

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PARI

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A368323 Expansion of e.g.f. exp(3*x) / (4 - 3*exp(x)).

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A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 3^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

A368322 Expansion of e.g.f. exp(2x) / (4 - 3exp(x)).

A368323 Expansion of e.g.f. exp(3x) / (4 - 3exp(x)).

A368324 Expansion of e.g.f. exp(4x) / (4 - 3exp(x)).