A032033 -id:A032033 - cp's OEIS frontend

A090353 G.f. satisfies A^4 = BINOMIAL(A^3).

1, 1, 4, 28, 286, 3886, 66260, 1361972, 32784353, 904412593, 28124223808, 973106096392, 37073604836768, 1541948625066176, 69513081435903392, 3376138396206853792, 175739519606046355540, 9760024269508314079444

Offset: 0

Paul D. Hanna, Nov 26 2003

In general, if A^n = BINOMIAL(A^(n-1)), then for all integer m>0 there exists an integer sequence B such that B^d = BINOMIAL(A^m) where d=gcd(m+1,n). Also, coefficients of A(k*x)^n = k-th binomial transform of coefficients in A(k*x)^(n-1) for all k>0.

			A^4 = BINOMIAL(A090355), since A090355=A^3. Also, BINOMIAL(A) = A090354^2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..320

Cf. A032033, A050352, A084784, A090351, A090354, A090355, A090356, A090358.

m:=40;
f:= func< n,x | Exp((&+[(&+[3^(j-1)*Factorial(j)* StirlingSecond(k,j)*x^k/k: j in [1..k]]): k in [1..n+2]])) >;
R:=PowerSeriesRing(Rationals(), m+1); // A090353
Coefficients(R!( f(m,x) )); // G. C. Greubel, Jun 09 2023

nmax = 17; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^4 - A[x/(1 - x)]^3/(1 - x) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)
With[{m=40}, CoefficientList[Series[Exp[Sum[Sum[3^(j-1)*j!* StirlingS2[k,j], {j,k}]*x^k/k, {k,m+1}]], {x,0,m}], x]] (* G. C. Greubel, Jun 09 2023 *)

{a(n) = my(A); if(n<0,0,A=1+x +x*O(x^n); for(k=1,n, B = subst(A^3,x,x/(1-x))/(1-x)+x*O(x^n); A = A - A^4 + B); polcoef(A,n,x))}
for(n=0,20,print1(a(n),", "))

m=50
def f(n, x): return exp(sum(sum(3^(j-1)*factorial(j)* stirling_number2(k,j)*x^k/k for j in range(1,k+1)) for k in range(1,n+2)))
def A090353_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(m,x) ).list()
A090353_list(m-9) # G. C. Greubel, Jun 09 2023

G.f. satisfies: A(x)^4 = A(x/(1-x))^3/(1-x).
a(n) ~ (n-1)! / (12 * (log(4/3))^(n+1)). - Vaclav Kotesovec, Nov 19 2014
O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*3^(k-1) = A050352(n) = 1/3*A032033(n) for n >= 1. - Peter Bala, May 26 2015
G.f. satisfies [x^n] 1/A(x)^(3*n-3) = [x^n] 1/A(x)^(4*n-4) for n >= 0. - Paul D. Hanna, Apr 28 2025
G.f.: Product_{k>=1} 1/(1 - k*x)^((1/12) * (3/4)^k). - Seiichi Manyama, May 26 2025

A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 1, -6, 6, 0, -1, 14, -36, 24, 0, 1, -30, 150, -240, 120, 0, -1, 62, -540, 1560, -1800, 720, 0, 1, -126, 1806, -8400, 16800, -15120, 5040, 0, -1, 254, -5796, 40824, -126000, 191520, -141120, 40320, 0, 1, -510, 18150, -186480, 834120, -1905120, 2328480, -1451520, 362880

Offset: 0

Peter Luschny, Jan 09 2017

sign

Signed version of A131689.

Integral_{x=0..1} F_n(x) = B_n(1) where B_n(x) are the Bernoulli polynomials.

			Triangle of coefficients starts:
[1]
[0,  1]
[0, -1,    2]
[0,  1,   -6,    6]
[0, -1,   14,  -36,    24]
[0,  1,  -30,  150,  -240,   120]
[0, -1,   62, -540,  1560, -1800,    720]
[0,  1, -126, 1806, -8400, 16800, -15120, 5040]

Peter Luschny, Illustration of the polynomials.
Peter Luschny, The Bernoulli Manifesto.
Grzegorz Rządkowski, Bernoulli numbers and solitons - revisited, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Row sums are A000012, diagonal is A000142.
Cf. A131689 (unsigned), A019538 (n>0, k>0), A090582.
Let F(n, x) = Sum_{k=0..n} T(n,k)*x^k then, apart from possible differences in the sign or the offset, we have: F(n, -5) = A094418(n), F(n, -4) = A094417(n), F(n, -3) = A032033(n), F(n, -2) = A004123(n), F(n, -1) = A000670(n), F(n, 0) = A000007(n), F(n, 1) = A000012(n), F(n, 2) = A000629(n), F(n, 3) = A201339(n), F(n, 4) = A201354(n), F(n, 5) = A201365(n).
Cf. A163626, A173018.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) - T(n-1, k))
end
for n in 0:7
    println([T(n,k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

F := (n,x) -> add((-1)^n*Stirling2(n,k)*k!*(-x)^k, k=0..n):
for n from 0 to 10 do PolynomialTools:-CoefficientList(F(n,x), x) od;

T[ n_, k_] := If[ n < 0 || k < 0, 0, (-1)^(n - k) k! StirlingS2[n, k]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(n + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

T(n, k) = (-1)^(n-k) * Stirling2(n, k) * k!.
E.g.f.: 1/(1-x*(1-exp(-t))) = Sum_{n>=0} F_n(x) t^n/n!.
T(n, k) = k*(T(n-1, k-1) - T(n-1, k)) for 0 <= k <= n, T(0, 0) = 1, otherwise 0.
Bernoulli numbers are given by B(n) = Sum_{k = 0..n} T(n, k) / (k+1) with B(1) = 1/2. - Michael Somos, Jul 08 2018
Let F_n(x) be the row polynomials of this sequence and W_n(x) the row polynomials of A163626. Then F_n(1 - x) = W_n(x) and Integral_{x=0..1} U(n, x) = Bernoulli(n, 1) for U in {W, F}. - Peter Luschny, Aug 10 2021
T(n, k) = [z^k] Sum_{k=0..n} Eulerian(n, k)*z^(k+1)*(z-1)^(n-k-1) for n >= 1, where Eulerian(n, k) = A173018(n, k). - Peter Luschny, Aug 15 2022

A090355 G.f. satisfies A^4 = BINOMIAL(A)^3.

Original entry on oeis.org

1, 3, 15, 109, 1086, 14178, 232906, 4647006, 109376595, 2967406345, 91130074437, 3123199831983, 118106517900868, 4883161763750820, 219076867059030300, 10597531747143624820, 549768536732090716371, 30443800514118532762329

Offset: 0

Paul D. Hanna, Nov 26 2003

See comments in A090353.

Cf. A090353, A090354; A019538, A032033, A050352, A201354.

Cf. A090352, A090357, A090362.

nmax = 17; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^4 - A[x/(1 - x)]^3/(1 - x)^3 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=local(A); if(n<1,0,A=1+x+x*O(x^n); for(k=1,n,B=subst(A,x,x/(1-x))/(1-x)+x*O(x^n); A=A-A^4+B^3);polcoeff(A,n,x))}

G.f.: A(x)^4 = A(x/(1-x))^3/(1-x)^3.
From Peter Bala, May 26 2015: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*3^k = A032033(n) = 3*A050352(n).
BINOMIAL(A(x)) = exp( Sum_{n >= 1} c(n)*x^n/n ) where c(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*4^k = A201354(n) = 4*A050352(n) for n >= 1. A(x) = B(x)^3 and BINOMIAL(A(x)) = B(x)^4 where B(x) = 1 + x + 4*x^2 + 28*x^3 + 286*x^4 + ... is the o.g.f. for A090353. See also A019538. (End)
G.f.: Product_{k>=1} 1/(1 - k*x)^((1/4) * (3/4)^k). - Seiichi Manyama, May 26 2025
a(n) ~ (n-1)! / (4 * log(4/3)^(n+1)). - Vaclav Kotesovec, May 28 2025

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

Original entry on oeis.org

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

A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 3, 15, 81, 489, 3237, 23211, 178707, 1467051, 12768345, 117263829, 1131901521, 11444383251, 120847326879, 1329303053391, 15197269729689, 180211641841353, 2212525627591533, 28078380387448515, 367782119667874083, 4965441830591976339, 69014083524412401873, 986364827548578356421

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A027710, A032033, A040027, A078940, A343523, A344735, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1 + 3 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 + 3 * x * A(x/(1 - x)) / (1 - x)^2.

A355111 Expansion of e.g.f. 3 / (4 - 3x - exp(3x)).

Original entry on oeis.org

1, 2, 11, 93, 1041, 14541, 243747, 4767183, 106556373, 2679469065, 74864397015, 2300883358995, 77144051804409, 2802027511061325, 109604157405491691, 4593512301562215783, 205348466229473678301, 9753645833118762303249, 490530576727430107027839, 26040317900991310393061499

Offset: 0

Ilya Gutkovskiy, Jun 19 2022

nonn

Cf. A003575, A006155, A032033, A255927, A343673, A355110, A355112, A355113, A355114.

nmax = 19; CoefficientList[Series[3/(4 - 3 x - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

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

a(n) ~ n! / ((1 + LambertW(exp(4))) * ((4 - LambertW(exp(4)))/3)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A365558 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(2/3).

Original entry on oeis.org

1, 2, 12, 112, 1432, 23272, 458952, 10644552, 283851272, 8555351112, 287585280392, 10666369505992, 432674936431112, 19054822031194952, 905387807689821832, 46166008179076287432, 2514469578906179506952, 145691888630159515550792

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A032033, A346982.

a[n_] := Sum[Product[3*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, 3*j+2)*stirling(n, k, 2));

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

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

A367472 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^2.

Original entry on oeis.org

1, 6, 60, 816, 13992, 289176, 6990360, 193432056, 6028092312, 208891033656, 7966989308760, 331618933474296, 14958464943057432, 726825458489514936, 37846457287387667160, 2102428978611587164536, 124109778776508893651352, 7758254465575303379273016

Offset: 0

Seiichi Manyama, Nov 19 2023

nonn

Cf. A032033, A367473.

Cf. A005649, A367470.

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

a(n) = Sum_{k=0..n} 3^k * (k+1)! * Stirling2(n,k).
a(0) = 1; a(n) = 3*Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

cp's OEIS Frontend

A090353 G.f. satisfies A^4 = BINOMIAL(A^3).

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A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k.

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A090355 G.f. satisfies A^4 = BINOMIAL(A)^3.

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

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Magma

Mathematica

PARI

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A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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Mathematica

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A355111 Expansion of e.g.f. 3 / (4 - 3*x - exp(3*x)).

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Mathematica

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A365558 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(2/3).

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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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Keywords

A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.

A355111 Expansion of e.g.f. 3 / (4 - 3x - exp(3x)).