Enrique Navarrete - cp's OEIS frontend

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

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)

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)

A387185 a(n) = n2^(n-1) + binomial(n,2)2^(n-2) + binomial(n,3)*2^(n-3).

Original entry on oeis.org

0, 1, 5, 19, 64, 200, 592, 1680, 4608, 12288, 32000, 81664, 204800, 505856, 1232896, 2969600, 7077888, 16711680, 39124992, 90898432, 209715200, 480772096, 1095761920, 2484076544, 5603590144, 12582912000, 28135391232, 62662901760, 139049566208, 307492814848, 677799526400

Offset: 0

Enrique Navarrete, Aug 21 2025

Number of ternary strings of length n that contain one, two or three 0's.

Number of words of length n defined on five letters that contain one a or 2 b's or 3 c's and any number of d's and e's.

			a(3) = 19 since the words are (number of permutations in parentheses): add (3), ade (6), aee (3), bbd (3), bbe (3), ccc (1).
a(4) = 64 since from the 81 strings of length 4 we subtract the following 17 (number of permutations in parentheses): 0000 (1), 1111 (1), 1112 (4), 1122 (6), 1222 (4), 2222 (1).

Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A001793, A385312.

a[n_] := Sum[2^(n-k)*Binomial[n, k], {k, 1, 3}]; Array[a, 30, 0] (* Amiram Eldar, Aug 21 2025 *)

E.g.f.: (1 + x/2 + x^2/6)*x*exp(2*x).
G.f.: x*(1-3*x+3*x^2)/(2*x-1)^4 . - R. J. Mathar, Aug 26 2025
a(n) = n*2^n*(20+3*n+n^2)/48. - R. J. Mathar, Aug 26 2025

A387083 Expansion of e.g.f. (2(1-x)^2)/(2-4x+x^2).

Original entry on oeis.org

1, 0, 1, 6, 42, 360, 3690, 44100, 602280, 9253440, 157966200, 2966317200, 60765843600, 1348539192000, 32229405608400, 825285553092000, 22541609025936000, 654175871661312000, 20101465198839024000, 651991603501798560000, 22260385752292527840000

Offset: 0

Enrique Navarrete, Aug 16 2025

For n > 0, a(n) is the number of ways to sit n people around circular tables, choose 2 people from each table, and linear order the tables.

			a(6)=3690 since for 6 people the number of ways to do the combined tasks in the comment are: 1800 using one table, 1080 using two tables with 4 and 2 people, 720 using two tables with 3 people each, and 90 using three tables with 2 people each.

Cf. A001286.

With[{m = 20}, CoefficientList[Series[(2*(1 - x)^2)/(2 - 4*x + x^2), {x, 0, m}], x] * Table[n!, {n, 0, m}]] (* Amiram Eldar, Aug 16 2025 *)

For n > 0, a(n) = ((1 + sqrt(2))^(n-1) - (sqrt(2) - 1)^(n-1)) * n! / 2^(n/2 + 1). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence 2*a(n) -4*n*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 26 2025
a(n) = A060995(n)*n!/2^n. - R. J. Mathar, Aug 26 2025

A387012 Number of ternary strings of length 2*n that have fewer 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 4, 48, 496, 4864, 46464, 436992, 4068096, 37601280, 345733120, 3166363648, 28910051328, 263320698880, 2393742770176, 21726260035584, 196938517118976, 1783247797223424, 16132649384411136, 145839570932465664, 1317564543167102976, 11896996193604993024, 107375816824319901696

Offset: 0

Enrique Navarrete, Aug 12 2025

			a(2) = 48 since the strings of length 4 are the following (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4), 1111 (1), 1112 (4), 1122 (6), 1222 (4), 2222 (1).

Cf. A001019, A128418, A385252.

a[n_] := 9^n - Sum[2^(n-k) * Binomial[2*n, n-k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Aug 16 2025 *)

a(n) = 9^n - Sum_{k=0..n} 2^(n-k)*binomial(2*n,n-k).
G.f.: (sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)-8*x*(1-9*x))/((1-9*x)*sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).
a(n) = A001019(n) - A128418(n).
D-finite with recurrence n*a(n) +(-29*n+28)*a(n-1) +12*(23*n-41)*a(n-2) +432*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

A386825 Triangle read by rows: T(n,k) = 3^(n-k)C(2n,n-k).

Original entry on oeis.org

1, 6, 1, 54, 12, 1, 540, 135, 18, 1, 5670, 1512, 252, 24, 1, 61236, 17010, 3240, 405, 30, 1, 673596, 192456, 40095, 5940, 594, 36, 1, 7505784, 2189187, 486486, 81081, 9828, 819, 42, 1, 84440070, 25019280, 5837832, 1061424, 147420, 15120, 1080, 48, 1, 956987460, 287096238

Offset: 0

Enrique Navarrete, Aug 04 2025

Row sums are A386826.

			Triangle begins:
        1;
        6,       1;
       54,      12,      1;
      540,     135,     18,     1;
     5670,    1512,    252,    24,    1;
    61236,   17010,   3240,   405,   30,   1;
   673596,  192456,  40095,  5940,  594,  36,  1;
  7505784, 2189187, 486486, 81081, 9828, 819, 42, 1;
  ...

Cf. A094527, A128417, A386826.

Flatten[Table[3^(n-k) Binomial[2n, n-k], {n, 0, 9}, {k, 0, n}]]

T(n,k) = 3^(n-k)*A094527(n,k).

A386826 a(n) = Sum_{k=0..n} 3^(n-k)C(2n,n-k).

Original entry on oeis.org

1, 7, 67, 694, 7459, 81922, 912718, 10273228, 116522275, 1329569290, 15244087642, 175472098996, 2026521318286, 23470106563924, 272476942589884, 3169997065488664, 36948020548661539, 431354994430077274, 5043279137171450914, 59041965004582271524, 692026745415822877594, 8119918150063503715324

Offset: 0

Enrique Navarrete, Aug 04 2025

Row sums of number triangle A386825.

Number of strings of length 2*n defined on {0,1,2,3} that have either the same number or more 0's than the combined number of 1's, 2's and 3's.

			a(3)=694 counts the strings of length 6 as follows: 540 strings with three 0's, 135 with four 0's, 18 with five 0's, and 1 string with six 0's. Hence 694 = 540 + 135 + 18 + 1, where the summands come from the triangle in A386825.

Cf. A128418, A386825.

Table[Sum[3^(n-k) Binomial[2n, n-k], {k, 0, n}], {n, 0, 21}]

a(n) = 3^n*binomial(2*n, n)*hypergeom([1, -n], [1+n], -1/3). - Stefano Spezia, Aug 05 2025
a(n) ~ 2^(2*n-1) * 3^(n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 07 2025
D-finite with recurrence n*a(n) +(n+1)*a(n-1) +2*(-946*n+2017)*a(n-2) +144*(286*n-821)*a(n-3) +122112*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Aug 26 2025
D-finite with recurrence n*(2*n-1)*a(n) +2*(-28*n^2+20*n+9)*a(n-1) +96*(2*n+1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 26 2025

A385252 Number of ternary strings of length 2*n that have at least one 0 but less 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 0, 32, 432, 4608, 45440, 432896, 4051712, 37535744, 345470976, 3165315072, 28905857024, 263303921664, 2393675661312, 21725991600128, 196937443377152, 1783243502256128, 16132632204541952, 145839502212988928, 1317564268289196032, 11896995094093365248, 107375812426273390592

Offset: 0

Enrique Navarrete, Jul 28 2025

nonn

			a(2)=32 since the strings of length 4 are (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4).
a(3)=432 since the strings of length 6 are (number of permutations in parentheses): 111110 (6), 111120 (30), 111220 (60), 112220 (60), 122220 (30), 222220 (6), 001111 (15), 001112 (60), 001122 (90), 001222 (60), 002222 (15).

Cf. A000302, A001019, A128418, A386670.

a[0]=0; a[n_]:=9^n - 4^n - Sum[2^(n-k)*Binomial[2n,n-k],{k,0,n}]; Array[a,22,0] (* Stefano Spezia, Jul 31 2025 *)

a(n) = 9^n - 4^n - Sum_{k=0..n} 2^(n-k)*C(2*n,n-k) for n > 0.
G.f.: (5*x*(sqrt(1-8*x))*(sqrt(1-8*x)+12*x-1)-8*x*(36*x^2-13*x+1))/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)*(36*x^2-13*x+1)) + 1.
a(n) = A001019(n) - A000302(n) - A128418(n), n > 0.
Conjecture D-finite with recurrence n*a(n) +(-37*n+36)*a(n-1) +4*(131*n-245)*a(n-2) +16*(-221*n+605)*a(n-3) +192*(59*n-213)*a(n-4) +6912*(-2*n+9)*a(n-5)=0. - R. J. Mathar, Jul 31 2025
a(n) = 9^n - 4^n - 2^n*binomial(2*n, n)*hypergeom([1, -n], [1+n], -1/2) for n > 0. - Stefano Spezia, Aug 05 2025

A386670 Number of ternary strings of length 2*n that have more 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 1, 9, 73, 577, 4521, 35313, 275577, 2150721, 16793929, 131230609, 1026283545, 8032614625, 62921342953, 493262044977, 3869724080313, 30379987189377, 238661880787593, 1876072096450257, 14756076838714713, 116126703647975457, 914363729294862633, 7203083947383222897

Offset: 0

Enrique Navarrete, Jul 28 2025

nonn

			a(1)=1 since the string of length 2 is 00.
a(2)=9 since the strings of length 4 are the 4 permutations of 0001, the 4 permutations of 0002, and 0000.
a(4)=577 since the strings of length 8 are (number of permutations in parentheses): 00000001 (8), 00000002 (8), 00000011 (28), 00000012 (56), 00000022 (28), 00000111 (56), 00000112 (168), 00000122 (168), 00000222 (56), 00000000 (1).

Cf. A054554, A059304, A128417, A128418, A243019.

a(n) = Sum_{k=1..n} 2^(n-k)*binomial(2*n,n-k).
a(n) = Sum_{k=1..n} A128417(n,k).
G.f.: (1-4*x-sqrt(1-8*x))/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).
a(n) = A128418(n) - A059304(n).

A386359 a(n) = (1/4)(9^n - 24^n + 3), n > 0; a(0) = 0.

Original entry on oeis.org

0, 1, 13, 151, 1513, 14251, 130813, 1187551, 10728913, 96724051, 871171813, 7843167751, 70598995513, 635432902651, 5719063896013, 51472246152751, 463252899729313, 4169286834982051, 37523624464511413, 337712791979294551, 3039415815008418313, 27354745083854834251

Offset: 0

Enrique Navarrete, Jul 19 2025

a(n) is the number of ternary strings of length 2*n with an even number of 1's and 2's (possibly zero) and a positive even number of 0's.

			For n=2, a(2)=13 since the strings of length 4 are (number of permutations in parentheses): 0022 (6), 0011 (6), 0000 (1).
For n=3, a(3)=151 since the strings of length 6 are (number of permutations in parentheses): 000000 (1), 000022 (15), 002222(15), 000011 (15), 001111 (15), 001122 (90).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (14,-49,36).

Cf. A386227, A386293.

I:=[0, 1, 13, 151]; [n le 4 select I[n] else 14*Self(n-1)-49*Self(n-2)+36*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 24 2025

LinearRecurrence[{14,-49,36},{0,1,13,151},23] (* Stefano Spezia, Jul 19 2025 *)

a(n) = 14*a(n-1) - 49*a(n-2) + 36*a(n-3), n > 3.
G.f.: (18*x^3 - x^2 + x)/((1 - 9*x)*(1 - 4*x)*(1 - x)).
E.g.f.: (1/4)*(exp(9*x) - 2*exp(4*x) + 3*exp(x) - 2).

cp's OEIS Frontend

User: Enrique Navarrete

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

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

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A387185 a(n) = n*2^(n-1) + binomial(n,2)*2^(n-2) + binomial(n,3)*2^(n-3).

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A387083 Expansion of e.g.f. (2*(1-x)^2)/(2-4*x+x^2).

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A387012 Number of ternary strings of length 2*n that have fewer 0's than the combined number of 1's and 2's.

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A386825 Triangle read by rows: T(n,k) = 3^(n-k)*C(2*n,n-k).

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A386826 a(n) = Sum_{k=0..n} 3^(n-k)*C(2*n,n-k).

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A385252 Number of ternary strings of length 2*n that have at least one 0 but less 0's than the combined number of 1's and 2's.

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A387185 a(n) = n2^(n-1) + binomial(n,2)2^(n-2) + binomial(n,3)*2^(n-3).

A387083 Expansion of e.g.f. (2(1-x)^2)/(2-4x+x^2).

A386825 Triangle read by rows: T(n,k) = 3^(n-k)C(2n,n-k).

A386826 a(n) = Sum_{k=0..n} 3^(n-k)C(2n,n-k).

A386359 a(n) = (1/4)(9^n - 24^n + 3), n > 0; a(0) = 0.