A362276 -id:A362276 - 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)!).

A362282 a(n) = n! * Sum_{k=0..floor(n/2)} (-n)^k * binomial(n-k,k)/(n-k)!.

Original entry on oeis.org

1, 1, -3, -17, 145, 1401, -19619, -267833, 5214273, 91975825, -2292948899, -49586832129, 1506939887377, 38595456391753, -1383612408628995, -40951481342092649, 1691614670048805121, 56809502720559644577, -2656760323700732460227, -99810124102484722532465

Offset: 0

Seiichi Manyama, Apr 14 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..394
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362276, A362277, A362281.

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

a(n) = A362277(n,2*n).
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)).

A362320 a(n) = n! * Sum_{k=0..floor(n/5)} (-n/5)^k / (k! * (n-5*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, -119, -863, -3527, -10751, -27215, 7197121, 96476689, 689534209, 3507486841, 14238448225, -5835497948279, -114117547235327, -1164586980639263, -8296447373407871, -46351121024513375, 25734702161134932481, 661538303263860440041

Offset: 0

Seiichi Manyama, Apr 16 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..434
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362276, A362304, A362315.

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

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

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

A362315 a(n) = n! * Sum_{k=0..floor(n/4)} (-n/4)^k /(k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, -23, -149, -539, -1469, 77281, 911737, 5657401, 25134121, -2065730039, -35352993389, -310739232803, -1913714425349, 213881558916481, 4797269708789041, 54560246286936241, 429606655679843857, -60718212515535701399, -1684610587476711352709

Offset: 0

Seiichi Manyama, Apr 15 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..431
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362276, A362304, A362320.

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

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

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

A362340 a(n) = n! * Sum_{k=0..floor(n/2)} (-k/2)^k / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 0, -2, 7, 51, -239, -2435, 16353, 209377, -1826099, -28232379, 303020125, 5494172893, -70032035163, -1457369472299, 21512472563281, 505400696581905, -8478758871011807, -221971772323923263, 4171251104170567101, 120416449897739144941

Offset: 0

Seiichi Manyama, Apr 17 2023

sign

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

Cf. A069856, A362341, A362342.

Cf. A362276.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^2/2))))

E.g.f.: exp(x) / (1 + LambertW(x^2/2)).

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)!.

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A362282 a(n) = n! * Sum_{k=0..floor(n/2)} (-n)^k * binomial(n-k,k)/(n-k)!.

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A362320 a(n) = n! * Sum_{k=0..floor(n/5)} (-n/5)^k / (k! * (n-5*k)!).

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A362315 a(n) = n! * Sum_{k=0..floor(n/4)} (-n/4)^k /(k! * (n-4*k)!).

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A362340 a(n) = n! * Sum_{k=0..floor(n/2)} (-k/2)^k / (k! * (n-2*k)!).

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