A361283 -id:A361283 - cp's OEIS frontend

A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 31, 73, 1, 1, 1, 9, 55, 241, 501, 1, 1, 1, 11, 85, 529, 2261, 4051, 1, 1, 1, 13, 121, 961, 6121, 24781, 37633, 1, 1, 1, 15, 163, 1561, 13041, 82711, 309835, 394353, 1, 1, 1, 17, 211, 2353, 24101, 207001, 1273567, 4342241, 4596553, 1

Offset: 0

Ilya Gutkovskiy, Sep 28 2017

			E.g.f. of column k: A_k(x) =  1 + x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 9*k + 1)*x^3/3! + (4*k^3 + 36*k^2 + 32*k + 1)*x^4/4! + ...
Square array begins:
  1,   1,    1,    1,     1,     1,  ...
  1,   1,    1,    1,     1,     1,  ...
  1,   3,    5,    7,     9,    11,  ...
  1,  13,   31,   55,    85,   121,  ...
  1,  73,  241,  529,   961,  1561,  ...
  1, 501, 2261, 6121, 13041, 24101,  ...

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

Columns k=0..4 give A000012, A000262, A082579, A091695, A361283.

Main diagonal gives A293013.

Table[Function[k, n! SeriesCoefficient[Exp[x/(1 - x)^k], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j-1, n-j)/j!); \\ Seiichi Manyama, Mar 06 2023

E.g.f. of column k: exp(x/(1 - x)^k).
From Seiichi Manyama, Oct 21 2017: (Start)
Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.
A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0. (End)
A(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j-1,n-j)/j!. - Seiichi Manyama, Mar 06 2023

A361280 Expansion of e.g.f. exp(x * (1+x)^4).

Original entry on oeis.org

1, 1, 9, 61, 481, 4881, 55321, 682669, 9343041, 139078081, 2216425321, 37736834301, 683184324769, 13064452686481, 262867726142841, 5549111222344621, 122499654278797441, 2819926900630750209, 67539541277010100681, 1679557316488693881661

Offset: 0

Seiichi Manyama, Mar 06 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..476

Column k=4 of A361277.

Cf. A361283.

A361280 := proc(n)
    n!*add(binomial(4*k,n-k)/k!,k=0..n) ;
end proc:
seq(A361280(n),n=0..60) ; # R. J. Mathar, Mar 12 2023

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^4)))

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(4, j-1)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..n} binomial(4*k,n-k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(4,k-1) * a(n-k)/(n-k)!.
D-finite with recurrence a(n) -a(n-1) +8*(-n+1)*a(n-2) -18*(n-1)*(n-2)*a(n-3) -16*(n-1)*(n-2)*(n-3)*a(n-4) -5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) ~ 5^(n/5 - 1/2) * n^(4*n/5) * exp(-256/15625 - 249*5^(4/5)*n^(1/5)/78125 + 236*5^(3/5)*n^(2/5)/9375 + 22*5^(2/5)*n^(3/5)/125 + 4*5^(-4/5)*n^(4/5) - 4*n/5) * (1 + 15409886*5^(1/5)/(48828125*n^(1/5))). - Vaclav Kotesovec, Nov 11 2023

A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

Original entry on oeis.org

1, 0, 2, 24, 252, 2880, 38280, 594720, 10565520, 209502720, 4558407840, 107702179200, 2744400415680, 75016089308160, 2189152249764480, 67906418407027200, 2230160988344889600, 77271779968704921600, 2815893910009609228800, 107629691727791474841600, 4304364116456244429388800

Offset: 0

Vaclav Kotesovec, Aug 24 2025

In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A293012, A000262, A082579, A091695, A361283.

Cf. A052845, A052887, A386514.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Aug 25 2025

nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]

For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).

A367790 E.g.f. satisfies A(x) = exp( x/(1-x)^4 * A(x) ).

Original entry on oeis.org

1, 1, 11, 148, 2669, 62056, 1777927, 60692920, 2408692505, 109074596320, 5553702114731, 314208715035304, 19561795753879909, 1329317730339826384, 97924919301787209647, 7773978186375852940696, 661702605336795904770353, 60119367618216155944350400

Offset: 0

Seiichi Manyama, Nov 30 2023

nonn

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

Cf. A052868, A362775, A367789.

Cf. A162475, A361283.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^4))))

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

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

A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

Original entry on oeis.org

1, 0, 2, 18, 156, 1560, 18480, 254520, 3973200, 68947200, 1312748640, 27175024800, 607314818880, 14566195163520, 373027570755840, 10154293067318400, 292659790712889600, 8899747730037964800, 284685195814757337600, 9553060139009702515200, 335468448755976164428800

Offset: 0

Enrique Navarrete, Aug 23 2025

For n > 0, a(n) is the number of ways to linearly order n distinguishable objects into one or several lines and then choose 2 objects from each line. If the lines are also linearly ordered see A364524.

A001804(n) is the number of ways if only 1 line is used.

			a(6)=18480 since there are 10800 ways using one line, 4320 ways with 2 lines using 2 and 4 objects, 3240 ways with 2 lines of 3 objects each, and 120 ways with 3 lines of 2 objects each.

Cf. A001804, A364524.
Cf. A293012, A000262, A082579, A091695, A361283.
Cf. A052845, A052887, A387244.

nmax = 20; CoefficientList[Series[E^(x^2/(1-x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* or *)
nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 3*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 24 2025 *)

From Vaclav Kotesovec, Aug 24 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 3*k-1) / k!.
a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-7)*a(n-2) + (n-2)*(n-1)*(4*n-11)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/8) * exp(1/27 - 3^(-5/4)*n^(1/4)/8 - 3^(-1/2)*n^(1/2)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n-1/8) / 2. (End)

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A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).

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A361280 Expansion of e.g.f. exp(x * (1+x)^4).

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A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

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A367790 E.g.f. satisfies A(x) = exp( x/(1-x)^4 * A(x) ).

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A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

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