A362478 -id:A362478 - cp's OEIS frontend

A362474 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)^2).

1, 1, 2, 10, 70, 646, 7576, 106744, 1761628, 33361948, 712950616, 16976294776, 445751093800, 12795850109992, 398697898011232, 13401365473319776, 483376669737381136, 18623161719254837008, 763300232417720682784, 33163224556779213475744

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..392
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=1 of A362483.
Cf. A362478, A362491.
Cf. A143768, A362475.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x^2/2*A[x]^2] + 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^2*exp(2*x))/2)))

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

A362490 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (3j+1)^(n-2j-1) / (j! * (n-3*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 17, 1, 1, 1, 1, 4, 33, 161, 1, 1, 1, 1, 5, 49, 321, 1351, 1, 1, 1, 1, 6, 65, 481, 2841, 12391, 1, 1, 1, 1, 7, 81, 641, 4471, 31641, 153385, 1, 1, 1, 1, 8, 97, 801, 6241, 57751, 498849, 2388905, 1

Offset: 0

Seiichi Manyama, Apr 22 2023

			Square array begins:
  1,    1,    1,    1,    1,    1,     1, ...
  1,    1,    1,    1,    1,    1,     1, ...
  1,    1,    1,    1,    1,    1,     1, ...
  1,    2,    3,    4,    5,    6,     7, ...
  1,   17,   33,   49,   65,   81,    97, ...
  1,  161,  321,  481,  641,  801,   961, ...
  1, 1351, 2841, 4471, 6241, 8151, 10201, ...

Winston de Greef, Table of n, a(n) for n = 0..11324 (150 antidiagonals)
Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A000012, A362477, A362478, A362479.

Cf. A362378.

T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(3*j+1)^(n-2*j-1)/(j!*(n-3*j)!));

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^3/6 * A_k(x)^3).
A_k(x) = exp(x - LambertW(-k*x^3/2 * exp(3*x))/3).
A_k(x) = ( -2 * LambertW(-k*x^3/2 * exp(3*x))/(k*x^3) )^(1/3) for k > 0.

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)!).

A362390 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)).

Original entry on oeis.org

1, 1, 1, 3, 17, 81, 441, 3641, 33825, 318753, 3505521, 45095601, 616484001, 9013086369, 145909533225, 2556431401161, 47388760825281, 937507626246081, 19840711661183457, 443937299529447009, 10456231167451597761, 259738234024404363201

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..434
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=2 of A362378.

Cf. A001470, A362478.

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

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

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

A362493 E.g.f. satisfies A(x) = exp(x - x^3/3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, -1, -31, -319, -2279, -4199, 269473, 7155233, 114846641, 920526641, -18415853279, -1115017249631, -31675298017271, -526379460621559, 2394778195929281, 603748739138745281, 27895091311964499553, 769764386129113157473, 6164705700089328588481

Offset: 0

Seiichi Manyama, Apr 22 2023

sign

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

Cf. A362492, A362494.

Cf. A362478.

my(N=30, 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)!).

cp's OEIS Frontend

A362474 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)^2).

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A362490 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (3*j+1)^(n-2*j-1) / (j! * (n-3*j)!).

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

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A362390 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)).

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A362493 E.g.f. satisfies A(x) = exp(x - x^3/3 * A(x)^3).

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A362490 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (3j+1)^(n-2j-1) / (j! * (n-3*j)!).