A362301 -id:A362301 - cp's OEIS frontend

A362043 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 * binomial(n-2j,j)/(n-2j)!.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 4, 9, 11, 1, 1, 1, 1, 5, 13, 21, 31, 1, 1, 1, 1, 6, 17, 31, 81, 106, 1, 1, 1, 1, 7, 21, 41, 151, 351, 337, 1, 1, 1, 1, 8, 25, 51, 241, 736, 1233, 1205, 1, 1, 1, 1, 9, 29, 61, 351, 1261, 2689, 5769, 5021, 1

Offset: 0

Seiichi Manyama, Apr 15 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,  5,  9,  13,  17,  21,  25, ...
  1, 11, 21,  31,  41,  51,  61, ...
  1, 31, 81, 151, 241, 351, 481, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000012, A190865, A001470.
Main diagonal gives A362173.
T(n,2*n) gives A362300.
T(n,6*n) gives A362301.
Cf. A359762, A362302.

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

E.g.f. of column k: exp(x + k*x^3/6).
T(n,k) = T(n-1,k) + k * binomial(n-1,2) * T(n-3,k) for n > 2.
T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j / (j! * (n-3*j)!).

A362173 a(n) = n! * Sum_{k=0..floor(n/3)} (n/6)^k * binomial(n-2k,k)/(n-2k)!.

Original entry on oeis.org

1, 1, 1, 4, 17, 51, 481, 3676, 18369, 272917, 3011201, 21058236, 427112401, 6160655359, 55380250017, 1423658493076, 25361574327041, 278603741558601, 8673295084155649, 183914415577719892, 2387417408385462801, 87273239189497636171, 2146479566819857007201

Offset: 0

Seiichi Manyama, Apr 14 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..441
Eric Weisstein's World of Mathematics, Lambert W-Function.

Main diagonal of A362043.

Cf. A277614, A362300, A362301.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-2*lambertw(-x^3/2))^(1/3))/(1+lambertw(-x^3/2))))

a(n) = n! * [x^n] exp(x + n*x^3/6).

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

A362305 a(n) = n! * Sum_{k=0..floor(n/3)} (-n)^k * binomial(n-2k,k)/(n-2k)!.

Original entry on oeis.org

1, 1, 1, -17, -95, -299, 12241, 122011, 642433, -41645015, -597247199, -4407324569, 390913189921, 7315513279933, 69439658097265, -7816418805235949, -180448412456686079, -2093964182367814319, 285679499679525805633, 7844019340520912230495

Offset: 0

Seiichi Manyama, Apr 15 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..402
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362301, A362302.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((lambertw(3*x^3)/3)^(1/3))/(1+lambertw(3*x^3))))

a(n) = A362302(n,6*n).
a(n) = n! * [x^n] exp(x - n*x^3).
E.g.f.: exp( ( LambertW(3*x^3)/3 )^(1/3) ) / (1 + LambertW(3*x^3)).

A362323 a(n) = n! * Sum_{k=0..floor(n/5)} n^k / (k! * (n-5*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 601, 4321, 17641, 53761, 136081, 181742401, 2415576241, 17245198081, 87699217321, 355981385761, 736792782125401, 14287010845685761, 145634558983324321, 1037210264169367681, 5794253172081059041, 16246379099801447769601

Offset: 0

Seiichi Manyama, Apr 16 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362301, A362321.

Cf. A190877, A362319.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-5*x^5)/5)^(1/5))/(1+lambertw(-5*x^5))))

a(n) = n! * [x^n] exp(x + n*x^5).

E.g.f.: exp( ( -LambertW(-5*x^5)/5 )^(1/5) ) / (1 + LambertW(-5*x^5)).

A362321 a(n) = n! * Sum_{k=0..floor(n/4)} n^k /(k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 97, 601, 2161, 5881, 1303681, 14723857, 90770401, 402581521, 139389608161, 2284512533161, 19946635524817, 122623661651401, 57728368477678081, 1240234284406887841, 14010634784751445441, 110117252571345122977

Offset: 0

Seiichi Manyama, Apr 16 2023

nonn

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

Cf. A362301, A362323.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-4*x^4)/4)^(1/4))/(1+lambertw(-4*x^4))))

a(n) = n! * [x^n] exp(x + n*x^4).

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

cp's OEIS Frontend

A362043 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 * binomial(n-2*j,j)/(n-2*j)!.

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A362173 a(n) = n! * Sum_{k=0..floor(n/3)} (n/6)^k * binomial(n-2*k,k)/(n-2*k)!.

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A362305 a(n) = n! * Sum_{k=0..floor(n/3)} (-n)^k * binomial(n-2*k,k)/(n-2*k)!.

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A362323 a(n) = n! * Sum_{k=0..floor(n/5)} n^k / (k! * (n-5*k)!).

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Programs

PARI

Formula

A362321 a(n) = n! * Sum_{k=0..floor(n/4)} n^k /(k! * (n-4*k)!).

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PARI

Formula

A362043 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 * binomial(n-2j,j)/(n-2j)!.

A362173 a(n) = n! * Sum_{k=0..floor(n/3)} (n/6)^k * binomial(n-2k,k)/(n-2k)!.

A362305 a(n) = n! * Sum_{k=0..floor(n/3)} (-n)^k * binomial(n-2k,k)/(n-2k)!.