A369803 -id:A369803 - 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

A369804 Expansion of 1/(1 - x^3/(1-x)^5).

Original entry on oeis.org

1, 0, 0, 1, 5, 15, 36, 80, 181, 431, 1060, 2617, 6401, 15521, 37513, 90741, 219918, 533619, 1295022, 3141826, 7619870, 18478155, 44810670, 108676262, 263576791, 639267800, 1550434777, 3760269946, 9119740067, 22118021213, 53642768716, 130099857234, 315531401964

Offset: 0

Seiichi Manyama, Feb 01 2024

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

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

Cf. A079675, A368475, A369803.
Cf. A369845, A369846, A369847, A369848.
Cf. A052920.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^3/(1-x)^5))

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

a(n) = A052920(5*n-3) for n > 0.
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/3)} binomial(n-1+2*k,n-3*k).
a(n) = A369845(n) - A369845(n-1). - R. J. Mathar, Feb 14 2024

A369840 Number of compositions of 5*n into parts 2 and 5.

Original entry on oeis.org

1, 1, 2, 7, 23, 68, 194, 555, 1601, 4633, 13404, 38752, 112004, 323728, 935737, 2704817, 7818464, 22599701, 65325542, 188826693, 545813094, 1577700612, 4560424135, 13182138184, 38103641048, 110140512968, 318366757185, 920255312908, 2660044812499, 7688994894381

Offset: 0

Seiichi Manyama, Feb 03 2024

nonn

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

Cf. A369803, A369842, A369843, A369844.
Cf. A099131, A369836, A369845.
Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {1, 1, 2, 7, 23}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A001687(5*n+1).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,n-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x^2).
a(n) = A369843(n) - A369843(n-1). - R. J. Mathar, Feb 14 2024

A369842 Number of compositions of 5*n-1 into parts 2 and 5.

Original entry on oeis.org

1, 3, 7, 18, 52, 154, 450, 1301, 3753, 10838, 31327, 90568, 261813, 756786, 2187496, 6323023, 18277014, 52830706, 152709940, 441415867, 1275934888, 3688154521, 10660798289, 30815580241, 89074003241, 257472939209, 744238632362, 2151259638423, 6218325456983

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,-9,10,-5,1).

Cf. A369803, A369840, A369843, A369844.

Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {1, 3, 7, 18, 52}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A001687(5*n).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+1+3*k,n-1-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^2/((1-x)^5 - x^2).

A369843 Number of compositions of 5*n-3 into parts 2 and 5.

Original entry on oeis.org

1, 2, 4, 11, 34, 102, 296, 851, 2452, 7085, 20489, 59241, 171245, 494973, 1430710, 4135527, 11953991, 34553692, 99879234, 288705927, 834519021, 2412219633, 6972643768, 20154781952, 58258423000, 168398935968, 486765693153, 1407021006061, 4067065818560

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,-9,10,-5,1).

Cf. A369803, A369840, A369842, A369844.

Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {1, 2, 4, 11, 34}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A001687(5*n-2).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,n-1-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x^2).

A369844 Number of compositions of 5*n-4 into parts 2 and 5.

Original entry on oeis.org

0, 1, 4, 11, 29, 81, 235, 685, 1986, 5739, 16577, 47904, 138472, 400285, 1157071, 3344567, 9667590, 27944604, 80775310, 233485250, 674901117, 1950836005, 5638990526, 16299788815, 47115369056, 136189372297, 393662311506, 1137900943868, 3289160582291, 9507486039274

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,-9,10,-5,1).

Cf. A369803, A369840, A369842, A369843.

Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {0, 1, 4, 11, 29}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A001687(5*n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+1+3*k,n-2-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^2*(1-x)/((1-x)^5 - x^2).

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

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A369804 Expansion of 1/(1 - x^3/(1-x)^5).

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A369840 Number of compositions of 5*n into parts 2 and 5.

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A369842 Number of compositions of 5*n-1 into parts 2 and 5.

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A369843 Number of compositions of 5*n-3 into parts 2 and 5.

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A369844 Number of compositions of 5*n-4 into parts 2 and 5.

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