A362043 -id:A362043 - cp's OEIS frontend

A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2k,k)/(n-2k)!.

1, 1, 1, 7, 33, 101, 1681, 14211, 72577, 1906633, 23242401, 166218911, 5966236321, 95016917997, 873707885233, 39767572858651, 781865428682241, 8787169718273681, 484500265577706817, 11335266937098816183, 150554918241183405601, 9749671976020428623221

Offset: 0

Seiichi Manyama, Apr 15 2023

nonn

Let k be a positive integer. It appears that reducing this sequence modulo k produces an eventually periodic sequence with period a multiple of k. For example, modulo 9 the sequence becomes [1, 1, 1, 7, 6, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, ...], with an apparent period [2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0] of length 18 starting at a(5). - Peter Bala, Apr 16 2023

Winston de Greef, Table of n, a(n) for n = 0..425
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362043, A362293.

Cf. A277614, A362314, A362319.

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

a(n) = A362043(n,2*n).
a(n) = n! * [x^n] exp(x + n*x^3/3).
E.g.f.: exp( ( -LambertW(-x^3) )^(1/3) ) / (1 + LambertW(-x^3)).
a(n) ~ (1 + 2*cos(2*Pi*mod(n,3)/3 - sqrt(3)/2)/exp(3/2)) * n^n / (sqrt(3) * exp(2*n/3 - 1)). - Vaclav Kotesovec, Apr 18 2023

A362302 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)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -1, -3, 1, 1, 1, 1, -2, -7, -9, 1, 1, 1, 1, -3, -11, -19, -9, 1, 1, 1, 1, -4, -15, -29, 1, 36, 1, 1, 1, 1, -5, -19, -39, 31, 211, 225, 1, 1, 1, 1, -6, -23, -49, 81, 526, 1009, 477, 1, 1, 1, 1, -7, -27, -59, 151, 981, 2353, 953, -819, 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,  0,  -1,  -2,  -3,  -4,  -5, ...
  1, -3,  -7, -11, -15, -19, -23, ...
  1, -9, -19, -29, -39, -49, -59, ...
  1, -9,   1,  31,  81, 151, 241, ...

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

Columns k=0..2 give A000012, A351929, A362309.
Main diagonal gives A362303.
T(n,2*n) gives A362304.
T(n,6*n) gives A362305.
Cf. A362043, A362277.

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)!).

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)!).

Original entry on oeis.org

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.

A362301 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, 19, 97, 301, 13681, 124951, 647809, 46543897, 612367201, 4447574011, 436897375201, 7505523945349, 70104150466897, 8735878156045951, 185209511009456641, 2114594302777738801, 319284313084581674689, 8053189772356178472547

Offset: 0

Seiichi Manyama, Apr 15 2023

nonn

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

Cf. A362043.

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) = A362043(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)).
a(n) ~ (1 + 2*cos(2*Pi*mod(n,3)/3 - 3^(1/6)/2)*exp(-3^(2/3)/2)) * 3^(n/3 - 1/2) * n^n / exp(2*n/3 - 1/3^(1/3)). - Vaclav Kotesovec, Apr 18 2023

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)).

cp's OEIS Frontend

A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2*k,k)/(n-2*k)!.

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A362302 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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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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PARI

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A362301 a(n) = n! * Sum_{k=0..floor(n/3)} n^k * binomial(n-2*k,k)/(n-2*k)!.

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Programs

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Formula

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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Crossrefs

Programs

PARI

Formula

A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2k,k)/(n-2k)!.

A362302 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)!.

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)!).

A362301 a(n) = n! * Sum_{k=0..floor(n/3)} n^k * binomial(n-2k,k)/(n-2k)!.

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