A362390 -id:A362390 - cp's OEIS frontend

A362378 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 * (j+1)^(n-2j-1) / (j! (n-3*j)!).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 9, 1, 1, 1, 1, 4, 17, 41, 1, 1, 1, 1, 5, 25, 81, 191, 1, 1, 1, 1, 6, 33, 121, 441, 1191, 1, 1, 1, 1, 7, 41, 161, 751, 3641, 9353, 1, 1, 1, 1, 8, 49, 201, 1121, 7351, 33825, 77897, 1

Offset: 0

Seiichi Manyama, Apr 20 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,   9,  17,  25,   33,   41,   49, ...
  1,  41,  81, 121,  161,  201,  241, ...
  1, 191, 441, 751, 1121, 1551, 2041, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A000012, A362381, A362390, A362391.

Cf. A362043, A362377.

T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(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)).
A_k(x) = exp(x - LambertW(-k*x^3/6 * exp(x))).
A_k(x) = -6 * LambertW(-k*x^3/6 * exp(x))/(k*x^3) for k > 0.

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

Original entry on oeis.org

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)!).

cp's OEIS Frontend

A362378 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 * (j+1)^(n-2j-1) / (j! (n-3*j)!).

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

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cp's OEIS Frontend

A362378 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 * (j+1)^(n-2*j-1) / (j! * (n-3*j)!).

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Author

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PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A362378 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 * (j+1)^(n-2j-1) / (j! (n-3*j)!).