A339014 -id:A339014 - cp's OEIS frontend

A194689 a(n) = Sum_{k=0..n} binomial(n,k)w(k)w(n-k) where w() = A000296().

1, 0, 2, 2, 14, 42, 222, 1066, 6078, 36490, 238046, 1653610, 12214270, 95361866, 784071966, 6764984362, 61066919230, 575200190986, 5640081557598, 57450510336234, 606773139773054, 6633515763375306, 74950634205257630, 873995513192234410, 10504736507220958142, 129983468625156713354

Offset: 0

N. J. A. Sloane, Sep 01 2011

nonn

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 771, Problem 37).

Seiichi Manyama, Table of n, a(n) for n = 0..560

Cf. A000296, A001861, A194689, A339014, A339017, A339027.

Table[Sum[(-1)^(n-k) * Binomial[n,k] * BellB[k,2] * 2^(n-k), {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jun 25 2022 *)

N=66;  x='x+O('x^N);
Q(k) = if (k>N, 1, 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ) );
gf=1/Q(0);  Vec(Ser(gf))
/* Joerg Arndt, Mar 07 2013 */

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1-x)))) \\ Seiichi Manyama, Nov 20 2020

G.f.: 1/Q(0) where Q(k) = 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
G.f.: 1/Q(0), where Q(k)= 1 - x*k - 2*x^2*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 06 2013
E.g.f.: exp(2*(exp(x) - 1 - x)). - Ilya Gutkovskiy, Apr 07 2018
a(0) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-1-k). - Seiichi Manyama, Nov 20 2020
a(n) ~ 4 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). - Vaclav Kotesovec, Jun 26 2022

A339017 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6)).

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 2, 2, 142, 506, 1346, 3170, 53198, 375234, 1880738, 7919082, 72104190, 678488362, 5164781154, 33220643026, 271431061614, 2710340281426, 26278673924322, 228727591600826, 2081516848032222, 21560234032116154, 236863265302626722, 2521687569105476002

Offset: 0

Ilya Gutkovskiy, Nov 19 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..569

Cf. A001861, A057837, A194689, A339014, A339027.

nmax = 27; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2 - x^3/6)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 27}]

my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1 - x - x^2/2 - x^3/6)))) \\ Michel Marcus, Nov 19 2020

a(0) = 1; a(n) = 2 * Sum_{k=4..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A057837(k) * A057837(n-k).

A339027 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6 - x^4 / 24)).

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 506, 1850, 5018, 12014, 26886, 1066782, 8193070, 42723722, 185108514, 719359762, 10426744914, 118490840686, 976376930502, 6583701431086, 38977418758494, 377188932759354, 4671829781287922, 51479602726372402, 483303800325691922

Offset: 0

Ilya Gutkovskiy, Nov 20 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..576

Cf. A001861, A057814, A194689, A339014, A339017.

nmax = 28; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2 - x^3/6 - x^4/24)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 5, n}]; Table[a[n], {n, 0, 28}]

my(x='x+O('x^30)); Vec(serlaplace(exp(2 * (exp(x) - 1 - x - x^2/2 - x^3/6 - x^4/24)))) \\ Michel Marcus, Nov 20 2020

a(0) = 1; a(n) = 2 * Sum_{k=5..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A057814(k) * A057814(n-k).

A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

Original entry on oeis.org

1, 4, 18, 90, 494, 2946, 18926, 130066, 950654, 7353794, 59954638, 513333618, 4601380766, 43062556322, 419742815726, 4252083713874, 44680229906622, 486145710591874, 5468499473222670, 63503107472489266, 760281866742088670, 9373065303624742498, 118858898763010225198

Offset: 0

Vaclav Kotesovec, Jun 25 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Cf. A000110, A001861, A035009, A194689, A217924, A293024, A339014.

nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+2*x], {x, 0, nmax}], x] * Range[0, nmax]!
Table[(BellB[n+2, 2] - BellB[n+1, 2])/4, {n, 0, 25}] (* Vaclav Kotesovec, Jul 21 2025 *)

a(n) ~ n^(n+2) * exp(n/LambertW(n/2) - n - 2) / (4 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+2)).

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k+1) * Bell(n-k+1). - Ilya Gutkovskiy, Jun 26 2022

A336345 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2 * (exp(x) - Sum_{j=0..k} x^j/j!)).

Original entry on oeis.org

1, 1, 2, 1, 0, 6, 1, 0, 2, 22, 1, 0, 0, 2, 94, 1, 0, 0, 2, 14, 454, 1, 0, 0, 0, 2, 42, 2430, 1, 0, 0, 0, 2, 2, 222, 14214, 1, 0, 0, 0, 0, 2, 42, 1066, 89918, 1, 0, 0, 0, 0, 2, 2, 142, 6078, 610182, 1, 0, 0, 0, 0, 0, 2, 2, 366, 36490, 4412798, 1, 0, 0, 0, 0, 0, 2, 2, 142, 3082, 238046, 33827974

Offset: 0

Seiichi Manyama, Nov 20 2020

			Square array begins:
     1,   1,  1, 1, 1, 1, 1, ...
     2,   0,  0, 0, 0, 0, 0, ...
     6,   2,  0, 0, 0, 0, 0, ...
    22,   2,  2, 0, 0, 0, 0, ...
    94,  14,  2, 2, 0, 0, 0, ...
   454,  42,  2, 2, 2, 0, 0, ...
  2430, 222, 42, 2, 2, 2, 0, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A001861, A194689, A339014, A339017, A339027.
Main diagonal gives A000007.
Cf. A293024.

{T(n, k) = n!*polcoef(prod(j=k+1, n, exp((x^j+x*O(x^n))/j!))^2, n)}

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << 2 * (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
  ary
end
def A336345(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A336345(20)

E.g.f. of column k: (Product_{j>k} exp(x^j/j!))^2.

T(0,k) = 1, T(1,k) = T(2,k) = ... = T(k,k) = 0 and T(n,k) = 2 * Sum_{j=k..n-1} binomial(n-1,j)*T(n-1-j,k) for n > k.

cp's OEIS Frontend

A194689 a(n) = Sum_{k=0..n} binomial(n,k)*w(k)*w(n-k) where w() = A000296().

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A339017 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6)).

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A339027 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6 - x^4 / 24)).

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A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

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A336345 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2 * (exp(x) - Sum_{j=0..k} x^j/j!)).

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Formula

A194689 a(n) = Sum_{k=0..n} binomial(n,k)w(k)w(n-k) where w() = A000296().