A362282 -id:A362282 - cp's OEIS frontend

A362277 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 * binomial(n-j,j)/(n-j)!.

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -2, 1, 1, 1, -3, -8, 1, 6, 1, 1, 1, -4, -11, 10, 41, 16, 1, 1, 1, -5, -14, 25, 106, 31, -20, 1, 1, 1, -6, -17, 46, 201, -44, -461, -132, 1, 1, 1, -7, -20, 73, 326, -299, -1952, -895, 28, 1, 1, 1, -8, -23, 106, 481, -824, -5123, -1028, 6481, 1216, 1

Offset: 0

Seiichi Manyama, Apr 13 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, -2, -5,  -8,  -11,  -14,   -17, ...
  1, -2,  1,  10,   25,   46,    73, ...
  1,  6, 41, 106,  201,  326,   481, ...
  1, 16, 31, -44, -299, -824, -1709, ...

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

Columns k=0..6 give A000012, (-1)^n * A001464(n), A293604, A362278, A362176, A362279, A362177.
Main diagonal gives A362276.
T(n,2*n) gives A362282.
Cf. A359762, A362302.

T(n,k) = n!*sum(j=0,n\2, (-k/2)^j/(j!*(n-2*j)!));

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

A362281 a(n) = n! * Sum_{k=0..floor(n/2)} n^k * binomial(n-k,k)/(n-k)!.

Original entry on oeis.org

1, 1, 5, 19, 241, 1601, 32581, 308995, 8655809, 106673761, 3805452901, 57704760851, 2500580809585, 45018720191329, 2295683481085541, 47848514992963651, 2806491306922172161, 66464103165835330625, 4407449313521981148229, 116893033842508769526931

Offset: 0

Seiichi Manyama, Apr 14 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..395
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277614, A362282.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(-lambertw(-2*x^2)/2))/(1+lambertw(-2*x^2))))

a(n) = n! * [x^n] exp(x + n*x^2).
E.g.f.: exp( sqrt( -LambertW(-2*x^2)/2 ) ) / (1 + LambertW(-2*x^2)).
a(n) ~ (1 + (-1)^n/exp(sqrt(2))) * 2^((n-1)/2) * n^n / exp(n/2 - 1/sqrt(2)). - Vaclav Kotesovec, Apr 15 2023

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

A362277 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 * binomial(n-j,j)/(n-j)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A362281 a(n) = n! * Sum_{k=0..floor(n/2)} n^k * binomial(n-k,k)/(n-k)!.

Views

Author

Keywords

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