A362397 -id:A362397 - cp's OEIS frontend

A362380 E.g.f. satisfies A(x) = exp(x + 3x^2/2 A(x)).

1, 1, 4, 19, 154, 1456, 18136, 260002, 4430812, 85170988, 1854422236, 44693165716, 1188169271488, 34434053438968, 1082632555160248, 36666259172292016, 1331754793762045456, 51622725829298301520, 2127683533625205288400

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

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

Column k=3 of A362377.

Cf. A362397.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + 3*x^2/2*A[x]] + 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(-3*x^2/2*exp(x)))))

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

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

A362394 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 * (j+1)^(n-j-1) / (j! * (n-2*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -5, 1, 1, 1, -2, -11, -14, 1, 1, 1, -3, -17, -11, 56, 1, 1, 1, -4, -23, 10, 381, 736, 1, 1, 1, -5, -29, 49, 976, 2461, 1114, 1, 1, 1, -6, -35, 106, 1841, 3736, -21083, -45156, 1, 1, 1, -7, -41, 181, 2976, 3121, -106910, -449623, -428660, 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,   0,   -1,   -2,   -3,   -4,    -5, ...
  1,  -5,  -11,  -17,  -23,  -29,   -35, ...
  1, -14,  -11,   10,   49,  106,   181, ...
  1,  56,  381,  976, 1841, 2976,  4381, ...
  1, 736, 2461, 3736, 3121, -824, -9539, ...

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

Columns k=0..3 give A000012, A362395, A362396, A362397.

Cf. A362277, A362377.

T(n, k) = n! * sum(j=0, n\2, (-k/2)^j*(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)).
A_k(x) = exp(x - LambertW(k*x^2/2 * exp(x))).
A_k(x) = 2 * LambertW(k*x^2/2 * exp(x))/(k*x^2) for k > 0.

cp's OEIS Frontend

A362380 E.g.f. satisfies A(x) = exp(x + 3x^2/2 A(x)).

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A362394 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 * (j+1)^(n-j-1) / (j! * (n-2*j)!).

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cp's OEIS Frontend

A362380 E.g.f. satisfies A(x) = exp(x + 3*x^2/2 * A(x)).

Views

Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

A362394 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 * (j+1)^(n-j-1) / (j! * (n-2*j)!).

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Examples

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Programs

PARI

Formula

A362380 E.g.f. satisfies A(x) = exp(x + 3x^2/2 A(x)).