A001804 -id:A001804 - cp's OEIS frontend

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

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)

A253284 Triangle read by rows, T(n,k) = (k+1)(n+1)!(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 2, 2, 6, 18, 12, 24, 144, 240, 120, 120, 1200, 3600, 4200, 1680, 720, 10800, 50400, 100800, 90720, 30240, 5040, 105840, 705600, 2116800, 3175200, 2328480, 665280, 40320, 1128960, 10160640, 42336000, 93139200, 111767040, 69189120, 17297280

Offset: 0

Peter Luschny, Mar 23 2015

G_n(x) = - Sum_{k=0..n} T(n,k)/(x-1)^(n+k+1) are generating functions, for n=0 of A000012, for n=1 of A002378, for n=2 of A083374 (with offset 0) and for n=3 for A253285. In general G_n(x) is the generating function of the sequence k -> ((n+k)!/k!)*C(n+k-1,k-1). These sequences are associated with the rows of the square array of unsigned Lah numbers (compare A253283 for the columns).

			Triangle begins:
1;
2, 2;
6, 18, 12;
24, 144, 240, 120;
120, 1200, 3600, 4200, 1680;
720, 10800, 50400, 100800, 90720, 30240;
5040, 105840, 705600, 2116800, 3175200, 2328480, 665280.

Cf. A000142, A001804, A001813, A002378, A006318, A083374, A088617, A253283, A253285.

/* As triangle: */ [[(k + 1)*Factorial(n + 1)*Factorial(n + k)/(Factorial(k + 1)^2*Factorial(n - k)): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 23 2015

T := (n,k) -> ((k+1)*(n+1)!*(n+k)!)/((k+1)!^2*(n-k)!);
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

f[n_] := Rest@ Flatten@ Reap@ Block[{i, k, t}, For[i = 0, i <= n, i++, For[k = 0, k <= i, k++, Sow[(i + 1)!*Binomial[i + k, i]*Binomial[i, k]/(k + 1)]]]]; f@ 7 (* Michael De Vlieger, Mar 23 2015 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1((n+1)!*binomial(n+k,n)*binomial(n,k)/(k+1), ", ");); print(););} \\ Michel Marcus, Mar 23 2015

T(n,k) = (n+1)!*binomial(n+k,n)*binomial(n,k)/(k+1).
T(n,k) = (n+1)!*A088617(n,k).
T(n,0) = n! = A000142(n).
T(n,1) = A001804(n+1) for n>0.
T(n,n) = (2*n)!/n! = A001813(n).
Sum_{k=0..n} T(n,k) = (n+1)!*hypergeom([-n, n+1], [2], -1) = (n+1)!*A006318(n).

A364524 Expansion of e.g.f. (1-x)^3/(1-3x+2x^2-x^3).

Original entry on oeis.org

1, 0, 2, 18, 168, 1920, 26640, 433440, 8064000, 168739200, 3922732800, 100310918400, 2798327347200, 84569169484800, 2752393009766400, 95978067913728000, 3569951024640000000, 141085196791418880000, 5903699220162551808000, 260764276201191899136000, 12124067910801279713280000

Offset: 0

Enrique Navarrete, Jul 27 2023

nonn

a(n) is the number of ways to seat n persons on benches, linear order the benches, and select 2 persons from each bench.

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

			a(6)=26640 since the number of ways to seat 6 persons on benches, linear order the benches, and select 2 persons from each bench is the following:
1) benches of type 1234,56: 1440 of these,  8640 ways;
2) benches of type 123,456:  720 of these,  6480 ways;
3) benches of type 12,34,56: 720 of these,   720 ways;
4) benches of type 123456:   720 of these, 10800 ways.

Cf. A001804, A002866.

With[{max = 20}, Range[0, max]! * CoefficientList[Series[(1 - x)^3/(1 - 3*x + 2*x^2 - x^3), {x, 0, max}], x]] (* Amiram Eldar, Jul 28 2023 *)

A387264 Expansion of e.g.f. exp(x^3/(1-x)^4).

Original entry on oeis.org

1, 0, 0, 6, 96, 1200, 14760, 196560, 2983680, 52315200, 1041465600, 22912243200, 545443113600, 13887294220800, 376188856243200, 10816657377926400, 329526966472704000, 10612556870243328000, 360307460991724646400, 12857257599818926694400, 480829913352068087808000

Offset: 0

Enrique Navarrete, Aug 24 2025

For n > 0, a(n) is the number of ways to seat n people on benches and select 3 people from each bench.

A001805 is the number of ways if only 1 bench is used.

			a(6)=14760 since there are 14400 ways using one bench and 360 ways with 2 benches of 3 people each.

Cf. A001804, A001805, A386514, A387244.

nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 4*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 25 2025 *)

From Vaclav Kotesovec, Aug 25 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 4*k-1)/k!.
a(n) = 5*(n-1)*a(n-1) - 10*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*(10*n-27)*a(n-3) - (n-3)*(n-2)*(n-1)*(5*n-21)*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(-27/1280 + 13*2^(-22/5)*n^(1/5)/25 + 13*2^(-19/5)*n^(2/5)/15 - 2^(-6/5)*n^(3/5) + 5*2^(-8/5)*n^(4/5) - n) * n^(n-1/10). (End)

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

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A253284 Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.

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A364524 Expansion of e.g.f. (1-x)^3/(1-3x+2x^2-x^3).

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A387264 Expansion of e.g.f. exp(x^3/(1-x)^4).

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A253284 Triangle read by rows, T(n,k) = (k+1)(n+1)!(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.