A362305 -id:A362305 - 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)!).

A362322 a(n) = n! * Sum_{k=0..floor(n/4)} (-n)^k / (k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, -95, -599, -2159, -5879, 1276801, 14669425, 90669601, 402407281, -136515598559, -2275742812199, -19922903656655, -122565283331399, 56538094207096321, 1235380139032068961, 13993348375743336001, 110062069784059565665

Offset: 0

Seiichi Manyama, Apr 16 2023

sign

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

Cf. A362282, A362305, A362324.

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)).

A362324 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, -599, -4319, -17639, -53759, -136079, 181137601, 2414356561, 17242917121, 87695201881, 355974659041, -734340892685399, -14279571631503359, -145614163414530719, -1037158816523518079, -5794132157196668639, 16192314610730781350401

Offset: 0

Seiichi Manyama, Apr 16 2023

sign

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

Cf. A362282, A362305, A362322.

Cf. A351906, A362320.

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)).

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)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A362322 a(n) = n! * Sum_{k=0..floor(n/4)} (-n)^k / (k! * (n-4*k)!).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362324 a(n) = n! * Sum_{k=0..floor(n/5)} (-n)^k / (k! * (n-5*k)!).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A362322 a(n) = n! * Sum_{k=0..floor(n/4)} (-n)^k / (k! * (n-4*k)!).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362324 a(n) = n! * Sum_{k=0..floor(n/5)} (-n)^k / (k! * (n-5*k)!).

Views

Author

Keywords

Links

Crossrefs

Programs

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)!.