A186021 -id:A186021 - cp's OEIS frontend

A332776 a(n) = 1 + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

1, 1, 2, 5, 18, 83, 464, 3041, 22810, 192595, 1807328, 18658097, 210138882, 2563990859, 33691089824, 474327797585, 7123141539610, 113656386574099, 1920170741071280, 34242622099969217, 642792206343361602, 12669617513914228907, 261613287097165614224, 5647565141926833774977

Offset: 0

Ilya Gutkovskiy, Jun 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..454

Cf. A000110, A006677, A007317, A080635, A105633, A186021.

a[n_] := a[n] = 1 + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
terms = 23; A[] = 0; Do[A[x] = Normal[Integrate[Exp[x] + A[x] (A[x] - 1), x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

E.g.f. A(x) satisfies: d/dx A(x) = exp(x) + A(x) * (A(x) - 1).
From Vaclav Kotesovec, Jun 09 2020: (Start)
E.g.f.: exp(x/2) * (BesselJ(2, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(2, 2*exp(x/2))) / (BesselJ(1, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(1, 2*exp(x/2))).
a(n) ~ n! / r^(n+1), where r = 1.0654335847261788612657252860730850911833168584... is the smallest real root of the equation BesselJ(1, 2*exp(r/2)) * BesselY(0,2) = BesselJ(0,2) * BesselY(1, 2*exp(r/2)). (End)

A282179 E.g.f.: exp(exp(x) - 1)(exp(3x) - 2*exp(x) + 1).

Original entry on oeis.org

0, 1, 9, 52, 283, 1561, 8930, 53411, 334785, 2199034, 15119621, 108644581, 814474176, 6358910949, 51615342685, 434865155292, 3796991928727, 34308796490005, 320379418256794, 3087939032182127, 30683582797977749, 313977721545709002, 3305220440084030809, 35759627532783842561

Offset: 0

Ilya Gutkovskiy, Feb 08 2017

nonn

Stirling transform of the cubes (A000578).

Exponential convolution of the sequences A000110 and A058481 (with a(0) = 0).

			E.g.f.: A(x) = x/1! + 9*x^2/2! + 52*x^3/3! + 283*x^4/4! + 1561*x^5/5! + 8930*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..572
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's MathWorld, Stirling Transform

Cf. A000110, A000578, A005494, A033452, A058481, A186021.

b:= proc(n, m) option remember; `if`(n=0,
       m^3, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 15 2022

Range[0, 23]! CoefficientList[Series[Exp[Exp[x] - 1] (Exp[3 x] - 2 Exp[x] + 1), {x, 0, 23}], x]
Table[Sum[StirlingS2[n, k] k^3, {k, 0, n}], {n, 0, 23}]
Table[Sum[Binomial[n, k] BellB[n-k] (3^k - 2), {k, 1, n}], {n, 0, 23}]
Table[BellB[n+3] - 3*BellB[n+2] + BellB[n], {n, 0, 23}] (* Vaclav Kotesovec, Aug 06 2021 *)

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000578(k).

a(n) = A000110(n) + A005494(n) - A186021(n+1).

A337186 a(n) = 1 + Sum_{k=0..n-2} binomial(n-2,k) * a(k).

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 36, 101, 308, 1013, 3562, 13300, 52482, 218045, 950614, 4335563, 20628882, 102153978, 525383324, 2801105889, 15455435864, 88117352141, 518391612686, 3142762585120, 19611454375090, 125829007917417, 829254498014570, 5608225148263459

Offset: 0

Ilya Gutkovskiy, Jan 29 2021

nonn

Cf. A000994, A007476, A186021.

a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]

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

A346771 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)) / (1 - x^2).

Original entry on oeis.org

1, 1, 1, 3, 7, 23, 81, 325, 1429, 6851, 35443, 196507, 1160633, 7266561, 48022313, 333776331, 2432140759, 18528143535, 147201596073, 1216952016245, 10448532393869, 92999784076875, 856739848236627, 8156691628658019, 80147320081510673, 811770418508099905

Offset: 0

Ilya Gutkovskiy, Aug 02 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..577

Cf. A000110, A000296, A040027, A060719, A186021, A344489.

nmax = 25; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)]/(1 - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 25; CoefficientList[Series[Exp[-x] (2 Exp[Exp[x] - 1] - 1), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(-x) * (2 * exp(exp(x) - 1) - 1).
a(0) = a(1) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A186021(k).
a(n) = 2 * A000296(n) - (-1)^n.

A352861 a(n) = 1 + Sum_{k=0..n-1} binomial(n+2,k+3) * a(k).

Original entry on oeis.org

1, 2, 7, 28, 121, 570, 2911, 15968, 93433, 580162, 3806275, 26284368, 190415809, 1442982350, 11409436363, 93913277608, 803094241309, 7121757279798, 65383520552131, 620517308328812, 6079168380979213, 61402851498255790, 638674759049919079, 6833589979500278700

Offset: 0

Ilya Gutkovskiy, Apr 06 2022

nonn

Cf. A000110, A032346, A032347, A045499, A045501, A186021, A352862.

a[n_] := a[n] = 1 + Sum[Binomial[n + 2, k + 3] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
nmax = 23; A[] = 0; Do[A[x] = 1/(1 - x) + x A[x/(1 - x)]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * A(x/(1 - x)) / (1 - x)^4.

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(n+1,k+2) * a(k).

A352862 a(n) = 1 + Sum_{k=0..n-1} binomial(n+3,k+4) * a(k).

Original entry on oeis.org

1, 2, 8, 36, 170, 865, 4742, 27757, 172375, 1130865, 7809057, 56572404, 428710587, 3389749264, 27901667938, 238599540142, 2115876327408, 19425465343555, 184355895494512, 1806122902809371, 18242807108024625, 189750478368293523, 2030261803964224359, 22323607721661782198

Offset: 0

Ilya Gutkovskiy, Apr 06 2022

nonn

Cf. A000110, A032346, A032347, A045499, A045500, A186021, A352861.

a[n_] := a[n] = 1 + Sum[Binomial[n + 3, k + 4] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
nmax = 23; A[] = 0; Do[A[x] = 1/(1 - x) + x A[x/(1 - x)]/(1 - x)^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

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

A257563 Triangle read by rows, coefficients T(n,k) of polynomials related to the Bell polynomials, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 1, 5, 4, 0, 1, 10, 14, 5, 0, 1, 19, 48, 30, 6, 0, 1, 36, 149, 158, 55, 7, 0, 1, 69, 445, 727, 413, 91, 8, 0, 1, 134, 1308, 3126, 2638, 924, 140, 9, 0, 1, 263, 3822, 12932, 15396, 7818, 1848, 204, 10, 0, 1, 520, 11159, 52278, 84920, 59382, 19998, 3396, 285, 11

Offset: 0

Peter Luschny, May 10 2015

The multivariate partial Bell polynomials as defined for example by L. Comtet, Advanced Combinatorics, depend on the variables x_1, x_2, ... Here we add a variable x_0 to cover the case of the first column (k=0) in the lower triangular matrix of the coefficients of the polynomials in a systematic way. The Bell polynomial is defined in the usual way as the row sum of the partial Bell polynomials and the univariate Bell polynomial as the Bell polynomial with all x_n set to the same variable x. We call these polynomials the x0-based univariate Bell polynomials. The classical univariate Bell polynomials are obtained by substituting x_0 = 0 prior to collapsing the x_n to one variable. In this case the coefficients of the polynomials are the Stirling subset numbers. The triangle given here are the coefficients of the polynomials in the general case.

			The triangle starts:
[1]
[0, 2]
[0, 1, 3]
[0, 1, 5, 4]
[0, 1, 10, 14, 5]
[0, 1, 19, 48, 30, 6]
[0, 1, 36, 149, 158, 55, 7]

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 135.

Peter Luschny, The Bell Transform

Cf. A000110, A048993, A186021.

T_row := proc(n) local T; T := proc(n,k) option remember; if k = 0 then x^n else add(binomial(n-1,j-1)*T(n-j,k-1)*x, j=0..n-k+1) fi end; PolynomialTools:-CoefficientList(add(T(n,k),k=0..n), x) end: seq(print(T_row(n)), n=0..6);

T[n_, k_] := T[n, k] = If[k==0, x^n, Sum[Binomial[n-1, j-1]*T[n-j, k-1]*x, {j, 0, n-k+1}]]; row[n_] := CoefficientList[Sum[T[n, k], {k, 0, n}], x]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 20 2016, adapted from Maple *)

def partial_bell_polynomial(n,k):
    X = var(['x_'+str(i) for i in (0..n+1)])
    @cached_function
    def T(n,k):
        if k==0: return X[0]^n
        return sum(binomial(n-1,j-1)*X[j]*T(n-j,k-1) for j in (0..n-k+1))
    return T(n,k).expand()
def univariate_bell_polynomial(n): # for comparison only
    p = sum(partial_bell_polynomial(n,k) for k in (0..n)).subs(x_0=0)
    q = p({p.variables()[i]:x for i in range(len(p.variables()))})
    R = PolynomialRing(QQ,'x')
    return R(q)
def x0_based_univariate_bell_polynomial(n):
    p = sum(partial_bell_polynomial(n,k) for k in (0..n))
    q = p({p.variables()[i]:x for i in range(len(p.variables()))})
    R = PolynomialRing(QQ,'x')
    return R(q)
for n in (0..6): x0_based_univariate_bell_polynomial(n).list()

Row sums are Bell(n)*(2-0^n) = A186021(n).

A333305 Irregular array read by rows, a refinement of A256894.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 3, 3, 1, 4, 3, 5, 1, 6, 1, 1, 1, 1, 4, 6, 4, 1, 5, 10, 9, 8, 7, 1, 10, 15, 9, 1, 10, 1, 1, 1, 1, 5, 10, 10, 5, 1, 6, 15, 14, 10, 35, 16, 15, 9, 1, 15, 60, 19, 15, 33, 12, 1, 20, 45, 14, 1, 15, 1, 1

Offset: 0

Peter Luschny, May 19 2020

			Irregular table (the refinement is indicated by round brackets) starts:
[0] [1]
[1] [1, 1]
[2] [1, (1, 1), 1]
[3] [1, (1, 2, 1), (3, 1), 1]
[4] [1, (1, 3, 3, 1), (4, 3, 5, 1), (6, 1), 1]
[5] [1, (1, 4, 6, 4, 1), (5, 10, 9, 8, 7, 1), (10, 15, 9, 1), (10, 1), 1]
[6] [1, (1, 5, 10, 10, 5, 1), (6, 15, 14, 10, 35, 16, 15, 9, 1), (15, 60, 19, 15,
     33, 12, 1), (20, 45, 14, 1), (15, 1), 1]

Peter Luschny, The Bell transform

Cf. A000070 (length of rows), A102356 (max in rows), A186021 (sum of rows).

Cf. A256894.

def BellBlocks(n):
    R = InfinitePolynomialRing(ZZ, 'v') # Thanks to F. Chapoton.
    V = R.gen()
    @cached_function
    def T(n, k):
        if k == 0: return V[0]^n
        return sum(binomial(n-1, j-1)*V[j]*T(n-j, k-1) for j in (0..n-k+1))
    P = (T(n, k) for k in (0..n))
    return flatten([p.coefficients() for p in P])
for n in (0..8): print(BellBlocks(n))

cp's OEIS Frontend

A332776 a(n) = 1 + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

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A282179 E.g.f.: exp(exp(x) - 1)*(exp(3*x) - 2*exp(x) + 1).

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A337186 a(n) = 1 + Sum_{k=0..n-2} binomial(n-2,k) * a(k).

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A346771 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)) / (1 - x^2).

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A352861 a(n) = 1 + Sum_{k=0..n-1} binomial(n+2,k+3) * a(k).

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A352862 a(n) = 1 + Sum_{k=0..n-1} binomial(n+3,k+4) * a(k).

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A257563 Triangle read by rows, coefficients T(n,k) of polynomials related to the Bell polynomials, for n>=0 and 0<=k<=n.

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A333305 Irregular array read by rows, a refinement of A256894.

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A282179 E.g.f.: exp(exp(x) - 1)(exp(3x) - 2*exp(x) + 1).