A362474 -id:A362474 - cp's OEIS frontend

A362478 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).

1, 1, 1, 3, 33, 321, 2841, 31641, 498849, 8979489, 167510961, 3427780401, 80374833441, 2089382321313, 58020408889353, 1721768971537161, 55150870311938241, 1897482353016075201, 69322763655015214689, 2676706914491568918369

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..399
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=2 of A362490.
Cf. A362474, A362491.
Cf. A362390.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x^3/3*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3*exp(3*x))/3)))

E.g.f.: exp(x - LambertW(-x^3 * exp(3*x))/3) = ( -LambertW(-x^3 * exp(3*x))/x^3 )^(1/3).

a(n) = n! * Sum_{k=0..floor(n/3)} (1/3)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A362491 E.g.f. satisfies A(x) = exp(x + x^4/4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 7, 151, 2251, 26251, 273841, 3281041, 61021801, 1518719401, 38199828151, 905801252071, 21398411003971, 560160675014851, 17260034904184801, 596005144436100001, 21359751419836426321, 773082506262449261521, 28839945213850125032551

Offset: 0

Seiichi Manyama, Apr 22 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362474, A362478.

Cf. A362473, A362494.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^4*exp(4*x))/4)))

E.g.f.: exp(x - LambertW(-x^4 * exp(4*x))/4) = ( -LambertW(-x^4 * exp(4*x))/x^4 )^(1/4).

a(n) = n! * Sum_{k=0..floor(n/4)} (1/4)^k * (4*k+1)^(n-3*k-1) / (k! * (n-4*k)!).

A365054 E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 64, 1038, 22666, 624448, 20801628, 813473468, 36543076444, 1854702411336, 104970490358944, 6555275229438664, 447773277245296536, 33211911279540910400, 2658266282912883209296, 228375288313274403201552, 20961681963345040127314192

Offset: 0

Seiichi Manyama, Aug 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..348
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A365053, A365055, A365056.

Cf. A362474, A362773.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*(1+x/2))/2)))

E.g.f.: exp( -LambertW(-2*x * (1+x/2))/2 ).
a(n) = n! * Sum_{k=0..n} (1/2)^(n-k) * (2*k+1)^(k-1) * binomial(k,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: sqrt(-LambertW(-2*x * (1+x/2)) / (2*x * (1+x/2))).
a(n) ~ sqrt((-sqrt(1 + exp(-1)) + 1 + exp(-1))/2) * n^(n-1) / (exp(n-1) * (-1 + sqrt(1 + exp(-1)))^n). (End)

A362475 E.g.f. satisfies A(x) = exp(x + 3x^2/2 A(x)^2).

Original entry on oeis.org

1, 1, 4, 28, 298, 4186, 74116, 1578340, 39394972, 1127378332, 36411516496, 1310173698736, 51982859674648, 2254757407407064, 106150698182657584, 5390926011965379376, 293782337188718257936, 17100576708082841577232, 1058920120014192744673600

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..370
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=3 of A362483.
Cf. A143768, A362474.
Cf. A362380.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x^2*exp(2*x))/2)))

E.g.f.: exp(x - LambertW(-3*x^2 * exp(2*x))/2) = sqrt( -LambertW(-3*x^2 * exp(2*x))/(3*x^2) ).

a(n) = n! * Sum_{k=0..floor(n/2)} (3/2)^k * (2*k+1)^(n-k-1) / (k! * (n-2*k)!).

A362483 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (2j+1)^(n-j-1) / (j! (n-2*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 10, 1, 1, 1, 4, 19, 70, 1, 1, 1, 5, 28, 169, 646, 1, 1, 1, 6, 37, 298, 2041, 7576, 1, 1, 1, 7, 46, 457, 4186, 30811, 106744, 1, 1, 1, 8, 55, 646, 7081, 74116, 560827, 1761628, 1, 1, 1, 9, 64, 865, 10726, 141901, 1578340, 11957905, 33361948, 1

Offset: 0

Seiichi Manyama, Apr 21 2023

			Square array begins:
  1,   1,    1,    1,    1,     1, ...
  1,   1,    1,    1,    1,     1, ...
  1,   2,    3,    4,    5,     6, ...
  1,  10,   19,   28,   37,    46, ...
  1,  70,  169,  298,  457,   646, ...
  1, 646, 2041, 4186, 7081, 10726, ...

Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A000012, A362474, A143768, A362475.

Cf. A362377, A362490.

T(n, k) = n! * sum(j=0, n\2, (k/2)^j*(2*j+1)^(n-j-1)/(j!*(n-2*j)!));

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^2/2 * A_k(x)^2).
A_k(x) = exp(x - LambertW(-k*x^2 * exp(2*x))/2).
A_k(x) = sqrt( -LambertW(-k*x^2 * exp(2*x))/(k*x^2) ) for k > 0.

cp's OEIS Frontend

A362478 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).

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A362491 E.g.f. satisfies A(x) = exp(x + x^4/4 * A(x)^4).

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A365054 E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x)^2 ).

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A362475 E.g.f. satisfies A(x) = exp(x + 3*x^2/2 * A(x)^2).

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A362483 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (2*j+1)^(n-j-1) / (j! * (n-2*j)!).

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A362475 E.g.f. satisfies A(x) = exp(x + 3x^2/2 A(x)^2).

A362483 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (2j+1)^(n-j-1) / (j! (n-2*j)!).