A362490 -id:A362490 - 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)!).

A362472 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, 7, 97, 961, 10201, 177241, 3801505, 80718625, 1887205681, 52896262321, 1648697978401, 54216677033377, 1928791931034697, 75326014326206281, 3159713152034201281, 140373558362282197441, 6632746205445950124385, 333591744669464008432225

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..384
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=6 of A362490.
Cf. A143768, A349562, A362473.
Cf. A362392, A362481.

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

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

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

A362477 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)^3).

Original entry on oeis.org

1, 1, 1, 2, 17, 161, 1351, 12391, 153385, 2388905, 40060781, 708351821, 13861042801, 305141790097, 7339275555067, 188198812659131, 5143808931521681, 150713978752271441, 4718460264313196665, 156524510548008965305, 5474266337362911068161

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..408
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=1 of A362490.

Cf. A362381.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3/2*exp(3*x))/3)))

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

a(n) = n! * Sum_{k=0..floor(n/3)} (1/6)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A362479 E.g.f. satisfies A(x) = exp(x + x^3/2 * A(x)^3).

Original entry on oeis.org

1, 1, 1, 4, 49, 481, 4471, 57751, 1036393, 19939753, 399150541, 9082285741, 237719388721, 6759766432849, 204408880370059, 6672899023062091, 236080878357745681, 8926817568378582481, 357421258163575234873, 15158257732928974255993

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..393
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=3 of A362490.

Cf. A362391.

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

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

a(n) = n! * Sum_{k=0..floor(n/3)} (1/2)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*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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A362472 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)^3).

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A362477 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)^3).

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A362479 E.g.f. satisfies A(x) = exp(x + x^3/2 * A(x)^3).

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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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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)!).