A362173 -id:A362173 - 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)!).

A362351 a(n) = n! * Sum_{k=0..floor(n/3)} (k/6)^k / (k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 61, 316, 1177, 11005, 84121, 434446, 5642781, 56725527, 374014005, 6211205456, 77331975281, 620174850521, 12539310726577, 186125334960730, 1757911008913141, 41887694462674691, 721886016954223661, 7846403629258814852, 215270385425700640905

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..469
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362350, A362352.

Cf. A362173.

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)).

a(n) ~ n^n * (exp(6^(1/3)*exp(-1/3)) + 2*cos(2^(-2/3)*3^(5/6)*exp(-1/3) - 2*Pi*n/3) / exp(2^(-2/3)*3^(1/3)*exp(-1/3))) / (2^(n/3) * 3^(n/3 + 1/2) * exp(2*n/3)). - Vaclav Kotesovec, Apr 18 2023

A362317 a(n) = n! * Sum_{k=0..floor(n/4)} (n/24)^k /(k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 5, 26, 91, 246, 2801, 26650, 159601, 702406, 12479941, 172561676, 1462655195, 8918930476, 215370384321, 3906667179836, 42828875064001, 333816101642140, 10190496077676901, 228789539391769336, 3077152545301687931, 29203537040556576776

Offset: 0

Seiichi Manyama, Apr 16 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..465
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362173, A362336.

Cf. A351932, A362314.

nmax = 30; CoefficientList[PowerExpand[Series[E^((-6*LambertW[-x^4/6])^(1/4)) / (1 + LambertW[-x^4/6]), {x, 0, nmax}]], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 18 2023 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-6*lambertw(-x^4/6))^(1/4))/(1+lambertw(-x^4/6))))

a(n) = n! * [x^n] exp(x + n*x^4/24).

E.g.f.: exp( ( -6*LambertW(-x^4/6) )^(1/4) ) / (1 + LambertW(-x^4/6)).

A362336 a(n) = n! * Sum_{k=0..floor(n/5)} (n/120)^k /(k! * (n-5*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 37, 148, 449, 1135, 15121, 172789, 1207009, 6106816, 24748725, 510855346, 8524169473, 84981641837, 602009065729, 3357322881625, 93871272204481, 2059974308136466, 26683062726210661, 243032907824598816, 1725747644222610625

Offset: 0

Seiichi Manyama, Apr 16 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..484
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362173, A362317.

Cf. A275423, A362319, A362323.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-24*lambertw(-x^5/24))^(1/5))/(1+lambertw(-x^5/24))))

a(n) = n! * [x^n] exp(x + n*x^5/120).

E.g.f.: exp( ( -24*LambertW(-x^5/24) )^(1/5) ) / (1 + LambertW(-x^5/24)).

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)!.

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A362351 a(n) = n! * Sum_{k=0..floor(n/3)} (k/6)^k / (k! * (n-3*k)!).

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A362317 a(n) = n! * Sum_{k=0..floor(n/4)} (n/24)^k /(k! * (n-4*k)!).

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A362336 a(n) = n! * Sum_{k=0..floor(n/5)} (n/120)^k /(k! * (n-5*k)!).

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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)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A362351 a(n) = n! * Sum_{k=0..floor(n/3)} (k/6)^k / (k! * (n-3*k)!).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362317 a(n) = n! * Sum_{k=0..floor(n/4)} (n/24)^k /(k! * (n-4*k)!).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A362336 a(n) = n! * Sum_{k=0..floor(n/5)} (n/120)^k /(k! * (n-5*k)!).

Views

Author

Keywords

Links

Crossrefs

Programs

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)!.