A351929 -id:A351929 - cp's OEIS frontend

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)!.

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)!).

A351930 Expansion of e.g.f. exp(x - x^4/24).

Original entry on oeis.org

1, 1, 1, 1, 0, -4, -14, -34, -34, 190, 1366, 5446, 11056, -30744, -421420, -2403764, -7434244, 9782396, 296347996, 2257819420, 9461601856, -1690329584, -395833164264, -3872875071064, -20629371958040, -17208144880024, 893208132927176, 10962683317693576

Offset: 0

Seiichi Manyama, Feb 26 2022

sign

Cf. A351929, A351931.

Cf. A351905, A351932.

a:= proc(n) option remember; `if`(n=0, 1,
     `if`(n<4, 0, -a(n-4)*binomial(n-1, 3))+a(n-1))
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, Feb 26 2022

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x-x^4/4!)))

a(n) = n!*sum(k=0, n\4, (-1/4!)^k*binomial(n-3*k, k)/(n-3*k)!);

a(n) = if(n<4, 1, a(n-1)-binomial(n-1, 3)*a(n-4));

a(n) = n! * Sum_{k=0..floor(n/4)} (-1/24)^k * binomial(n-3*k,k)/(n-3*k)!.

D-finite with recurrence a(n) = a(n-1) - binomial(n-1,3) * a(n-4) for n > 3.

A351931 Expansion of e.g.f. exp(x - x^5/120).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, -5, -20, -55, -125, -125, 925, 7525, 34750, 124125, 249250, -1013375, -14708875, -97413875, -477236375, -1443329375, 3466472500, 91499089375, 804081585000, 5030009685625, 20366827624375, -23484049500625, -1391395435656875, -15503027252406875

Offset: 0

Seiichi Manyama, Feb 26 2022

sign

Cf. A351929, A351930.

Cf. A275423, A351906.

m = 28; Range[0, m]! * CoefficientList[Series[Exp[x - x^5/5!], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x-x^5/5!)))

a(n) = n!*sum(k=0, n\5, (-1/5!)^k*binomial(n-4*k, k)/(n-4*k)!);

a(n) = if(n<5, 1, a(n-1)-binomial(n-1, 4)*a(n-5));

a(n) = n! * Sum_{k=0..floor(n/5)} (-1/5!)^k * binomial(n-4*k,k)/(n-4*k)!.

a(n) = a(n-1) - binomial(n-1,4) * a(n-5) for n > 4.

A362341 a(n) = n! * Sum_{k=0..floor(n/3)} (-k/6)^k / (k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, 0, -3, -9, 21, 246, 1065, -4283, -67319, -397484, 2315941, 45914155, 343743037, -2623221054, -62980998639, -571382718039, 5391435590545, 152175023203432, 1622112809355661, -18232162910685569, -591788241447761819, -7247966654986009490

Offset: 0

Seiichi Manyama, Apr 17 2023

sign

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

Cf. A069856, A362340, A362342.

Cf. A351929, A362303, A362348.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^3/6))))

E.g.f.: exp(x) / (1 + LambertW(x^3/6)).

cp's OEIS Frontend

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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A351930 Expansion of e.g.f. exp(x - x^4/24).

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Maple

PARI

PARI

PARI

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A351931 Expansion of e.g.f. exp(x - x^5/120).

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Mathematica

PARI

PARI

PARI

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A362341 a(n) = n! * Sum_{k=0..floor(n/3)} (-k/6)^k / (k! * (n-3*k)!).

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PARI

Formula

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)!.