A369804 -id:A369804 - cp's OEIS frontend

A368475 Expansion of o.g.f. (1-x)^5/((1-x)^5 - x^4).

1, 0, 0, 0, 1, 5, 15, 35, 71, 136, 265, 550, 1211, 2732, 6126, 13485, 29191, 62648, 134408, 289656, 627401, 1363124, 2963186, 6434484, 13951852, 30221185, 65442625, 141745045, 307137901, 665732417, 1443184210, 3128438335, 6780867186, 14696002913, 31848721632

Offset: 0

Enrique Navarrete, Dec 26 2023

For n > 0, a(n) is the number of ways to split [n] into an unspecified number of intervals and then choose 4 blocks (i.e., subintervals) from each interval. For example, for n=12, a(12)=1211 since the number of ways to split [12] into intervals and then select 4 blocks from each interval is C(12,4) + C(8,4)*C(4,4) + C(7,4)*C(5,4) + C(6,4)*C(6,4) + C(5,4)*C(7,4) + C(4,4)*C(8,4) + C(4,4)*C(4,4)*C(4,4) for a total of 1211 ways.
For n > 0, a(n) is also the number of compositions of n using parts of size at least 4 where there are binomial(i,4) types of i, i >= 4 (see example).
Number of compositions of 5*n-4 into parts 4 and 5. - Seiichi Manyama, Feb 01 2024

			Since there are C(4,4) = 1 type of 4, C(5,4) = 5 types of 5, C(6,4) = 15 types of 6, C(7,4) = 35 types of 7, C(8,4) = 70 types of 8, and (12,4) = 495 types of 12, we can write 12 in the following ways:
  12: 495 ways;
  8+4: 70 ways;
  7+5: 175 ways;
  6+6: 225 ways;
  5+7: 175 ways;
  4+8: 70 ways;
  4+4+4: 1 way, for a total of 1211 ways.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-4,1).

Cf. A000332, A017827, A099131, A290998.
Cf. A079675, A369803, A369804.
Cf. A088305, A095263, A290998, A369794, A369809.

CoefficientList[Series[(1 - x)^5/((1 - x)^5 - x^4), {x, 0, 50}], x] (* Wesley Ivan Hurt, Dec 26 2023 *)

Vec((1-x)^5/((1-x)^5 - x^4) + O(x^40)) \\ Michel Marcus, Dec 27 2023

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5), n>=6; a(0)=1, a(1)=a(2)=a(3)=0, a(4)=1, a(5)=5.
G.f.: 1/(1-Sum_{k>=4} binomial(k,4)*x^k).
G.f.: 1/p(S), where p(S) = 1 - S^4 - S^5 and S = x/(1-x).
First differences of A099131. - R. J. Mathar, Jan 29 2024
a(n) = A017827(5*n-4) = Sum_{k=0..floor((5*n-4)/4)} binomial(k,5*n-4-4*k) for n > 0. - Seiichi Manyama, Feb 01 2024
a(n) = Sum_{k=0..floor(n/4)} binomial(n-1+k,n-4*k). - Seiichi Manyama, Feb 02 2024

A369845 Number of compositions of 5*n into parts 3 and 5.

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 58, 138, 319, 750, 1810, 4427, 10828, 26349, 63862, 154603, 374521, 908140, 2203162, 5344988, 12964858, 31443013, 76253683, 184929945, 448506736, 1087774536, 2638209313, 6398479259, 15518219326, 37636240539, 91279009255, 221378866489

Offset: 0

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-5,1).

Cf. A369804, A369846, A369847, A369848.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {1, 1, 1, 2, 7}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+2*k, n-3*k));

a(n) = A052920(5*n).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,n-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x^3).
a(n) = A369847(n+1)-A369847(n). - R. J. Mathar, Feb 14 2024

A369803 Expansion of 1/(1 - x^2/(1-x)^5).

Original entry on oeis.org

1, 0, 1, 5, 16, 45, 126, 361, 1046, 3032, 8771, 25348, 73252, 211724, 612009, 1769080, 5113647, 14781237, 42725841, 123501151, 356986401, 1031887518, 2982723523, 8621714049, 24921502864, 72036871920, 208226244217, 601888555723, 1739789499591, 5028950081882

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Number of compositions of 5*n-2 into parts 2 and 5.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,10,-5,1).

Cf. A079675, A368475, A369804.
Cf. A369840, A369842, A369843, A369844.
Cf. A001687.

my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^5))

a(n) = sum(k=0, n\2, binomial(n-1+3*k, n-2*k));

a(n) = A001687(5*n-1) for n > 0.
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+3*k,n-2*k).
a(n) = A369840(n)-A369840(n-1). - R. J. Mathar, Feb 14 2024

A369846 Number of compositions of 5*n-1 into parts 3 and 5.

Original entry on oeis.org

0, 1, 4, 10, 21, 44, 101, 250, 629, 1557, 3784, 9120, 21992, 53228, 129177, 313701, 761403, 1846804, 4478044, 10858285, 26332515, 63865592, 154900529, 375691009, 911166977, 2209835169, 5359470121, 12998281146, 31524747503, 76457088518, 185431544730

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-5,1).

Cf. A369804, A369845, A369847, A369848.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {0, 1, 4, 10, 21}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+1+2*k, n-2-3*k));

a(n) = A052920(5*n-1).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+1+2*k,n-2-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^2*(1-x)/((1-x)^5 - x^3).

A369847 Number of compositions of 5*n-2 into parts 3 and 5.

Original entry on oeis.org

1, 2, 3, 5, 12, 34, 92, 230, 549, 1299, 3109, 7536, 18364, 44713, 108575, 263178, 637699, 1545839, 3749001, 9093989, 22058847, 53501860, 129755543, 314685488, 763192224, 1850966760, 4489176073, 10887655332, 26405874658, 64042115197, 155321124452

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-5,1).

Cf. A369804, A369845, A369846, A369848.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {1, 2, 3, 5, 12}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+2*k, n-1-3*k));

a(n) = A052920(5*n-2).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,n-1-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x^3).

A369848 Number of compositions of 5*n-4 into parts 3 and 5.

Original entry on oeis.org

0, 1, 3, 6, 11, 23, 57, 149, 379, 928, 2227, 5336, 12872, 31236, 75949, 184524, 447702, 1085401, 2631240, 6380241, 15474230, 37533077, 91034937, 220790480, 535475968, 1298668192, 3149634952, 7638811025, 18526466357, 44932341015, 108974456212, 264295580664

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-5,1).

Cf. A369804, A369845, A369846, A369847.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {0, 1, 3, 6, 11}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+2*k, n-2-3*k));

a(n) = A052920(5*n-4).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,n-2-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^2*(1-x)^2/((1-x)^5 - x^3).

A369686 LCM-transform of A359804 (see Comment and links).

Original entry on oeis.org

1, 2, 3, 5, 2, 1, 1, 7, 3, 2, 1, 1, 11, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 13, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 17, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 11, 1, 5, 1, 1, 1, 1, 1, 1, 3, 31, 1

Offset: 1

David James Sycamore, Jan 28 2024

nonn

Let b(k) be the Least Common Multiple (LCM) of the first k terms of A359804, then a(n) = b(n)/b(n-1), where sequence b(n) is A369685.

The property S (as defined in A368900) refers to what is observed in the positive integers (A000027), and also in the Doudna sequence (A005940), whereby each prime power appears prior to any of its multiples. The present sequence does not have this property since, for example, 26 = a(31) precedes 13 = a(42). Thus A369804 represents a significant disturbance of A000027 in that whereas it is conjectured to be a permutation of the positive integers, it does not preserve one of the basic properties of that sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Andrzej Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
Andrzej Nowicki, Strong divisibility and LCM-sequences, Am. Math. Monthly (2015) Vol. 122, 958-966.

Cf. A014963, A359804, A368900, A369685.

nn = 120; c[] = False; q[] = 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
Set[{i, j}, {1, 2}]; m = 2; u = 3;
Do[
  (k = q[#]; While[c[k #], k++]; k *= #; While[c[# q[#]], q[#]++]) &[
  (p = 2; While[Divisible[i j, p], p = NextPrime[p]]; p)];
  Set[{a[n], c[k], i, j, m}, {#/m, True, j, k, #}] &[LCM[m, k]];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jan 29 2024 *)

a(n) = A369685(n)/A369685(n-1).

More terms from Michael De Vlieger, Jan 29 2024

cp's OEIS Frontend

A368475 Expansion of o.g.f. (1-x)^5/((1-x)^5 - x^4).

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A369845 Number of compositions of 5*n into parts 3 and 5.

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A369803 Expansion of 1/(1 - x^2/(1-x)^5).

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A369846 Number of compositions of 5*n-1 into parts 3 and 5.

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A369847 Number of compositions of 5*n-2 into parts 3 and 5.

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A369848 Number of compositions of 5*n-4 into parts 3 and 5.

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A369686 LCM-transform of A359804 (see Comment and links).

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