A099131 -id:A099131 - cp's OEIS frontend

A373912 Number of compositions of 7*n into parts 6 and 7.

1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 3448, 6556, 12121, 22509, 43453, 89150, 193823, 436304, 989759, 2219064, 4869285, 10434412, 21900170, 45297211, 93054446, 191371581, 396480142, 830227401, 1756883373, 3746468095, 8017653633, 17151612398

Offset: 0

Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-6,1).

Cf. A373907, A373908, A373909, A373910, A373911.
Cf. A099099, A099131, A107025, A373913.
Cf. A017847.

a(n) = sum(k=0, n\6, binomial(n+k, n-6*k));

a(n) = A017847(7*n).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^6/(1 - x)^6).

A107025 Binomial transform of the expansion of 1/(1-x^5-x^6).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 938, 1808, 3459, 6826, 14198, 30960, 69143, 154433, 340006, 734561, 1561313, 3286129, 6900097, 14542101, 30855957, 65908862, 141395972, 303745077, 651763377, 1395140215, 2978858672

Offset: 0

Paul Barry, May 09 2005

In general, the binomial transform of 1/(1-x^r-x^(r+1)) is given by (1-x)^r/((1-x)^(r+1)-x^r), with a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+k,(r+1)k) = Sum_{k=0..floor((r+1)n/r)} binomial(k,(r+1)n-r*k).

Number of compositions of 6*n into parts 5 and 6. - Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A017837, A099099, A099131.

G.f.: (1-x)^5/((1-x)^6-x^5).
a(n) = 6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+7a(n-5)-a(n-6).
a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+k, 6k).
a(n) = Sum_{k=0..floor(6n/5)} binomial(k, 6n-5k).
a(n) = A017837(6*n). - Seiichi Manyama, Jun 22 2024

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

Original entry on oeis.org

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

A369836 Number of compositions of 5*n into parts 1 and 5.

Original entry on oeis.org

1, 2, 8, 34, 140, 571, 2328, 9496, 38740, 158045, 644761, 2630364, 10730820, 43777405, 178594110, 728591751, 2972359720, 12126025705, 49469281395, 201814663875, 823322219501, 3358821723401, 13702634402876, 55901207340276, 228054320813276, 930369409108152

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 (6,-10,10,-5,1).

Cf. A079675, A369837, A369838, A369839.
Cf. A099131, A369840, A369845.
Cf. A003520.

LinearRecurrence[{6, -10, 10, -5, 1}, {1, 2, 8, 34, 140}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A003520(5*n).
a(n) = Sum_{k=0..n} binomial(n+4*k,n-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x).

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

A369849 Number of compositions of 5*n-1 into parts 4 and 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 13, 35, 92, 220, 484, 1013, 2092, 4382, 9404, 20552, 45185, 99009, 215481, 466361, 1006897, 2174834, 4705895, 10200142, 22128873, 48009456, 104111224, 225655617, 488945055, 1059372394, 2295532150, 4974876116, 10782658417, 23371307904, 50655960304

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,-10,10,-4,1).

Cf. A099131, A368475, A369850, A369851.

Cf. A017827.

LinearRecurrence[{5, -10, 10, -4, 1}, {1, 2, 3, 4, 6}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\4, binomial(n+k, n-1-4*k));

a(n) = A017827(5*n-1).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-1-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x^4).

A369850 Number of compositions of 5*n-2 into parts 4 and 5.

Original entry on oeis.org

0, 1, 3, 6, 10, 16, 29, 64, 156, 376, 860, 1873, 3965, 8347, 17751, 38303, 83488, 182497, 397978, 864339, 1871236, 4046070, 8751965, 18952107, 41080980, 89090436, 193201660, 418857277, 907802332, 1967174726, 4262706876, 9237582992, 20020241409, 43391549313

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,-10,10,-4,1).

Cf. A099131, A368475, A369849, A369851.

Cf. A017827.

LinearRecurrence[{5, -10, 10, -4, 1}, {0, 1, 3, 6, 10}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\4, binomial(n+k, n-2-4*k));

a(n) = A017827(5*n-2).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-2-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x^2*(1-x)^2/((1-x)^5 - x^4).

A369851 Number of compositions of 5*n-3 into parts 4 and 5.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 36, 65, 129, 285, 661, 1521, 3394, 7359, 15706, 33457, 71760, 155248, 337745, 735723, 1600062, 3471298, 7517368, 16269333, 35221440, 76302420, 165392856, 358594516, 777451793, 1685254125, 3652428851, 7915135727, 17152718719, 37172960128

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,-10,10,-4,1).

Cf. A099131, A368475, A369849, A369850.

Cf. A017827.

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

a(n) = sum(k=0, n\4, binomial(n+k, n-3-4*k));

a(n) = A017827(5*n-3).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-3-4*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5).
G.f.: x^3*(1-x)/((1-x)^5 - x^4).

A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2k,n-4k).

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 29, 85, 212, 476, 1016, 2172, 4825, 11213, 26763, 64095, 151851, 354737, 820328, 1889968, 4361521, 10106859, 23509678, 54793282, 127709888, 297336790, 691382201, 1606284377, 3731020629, 8668253125, 20146856893, 46840732201, 108918637566, 253262275888

Offset: 0

Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-14,6,-1).

Cf. A003269, A003522, A005252, A038503, A099131.

Cf. A107025, A373904, A373905.

a(n) = sum(k=0, n\4, binomial(n+2*k, n-4*k));

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 14*a(n-4) + 6*a(n-5) - a(n-6).

G.f.: 1/(1 - x - x^4/(1 - x)^5).

A373913 Number of compositions of 8*n into parts 7 and 8.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 10, 46, 166, 496, 1288, 3004, 6437, 12888, 24464, 44728, 80428, 146320, 278104, 564929, 1225811, 2778772, 6396236, 14620646, 32760586, 71565796, 152344397, 316911454, 647536777, 1308456096, 2635130392, 5330198752, 10896635912

Offset: 0

Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,9,-1).

Cf. A099099, A099131, A107025, A373912.

Cf. A017857.

CoefficientList[Series[1/(1-x-x^7/(1-x)^7),{x,0,40}],x] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,9,-1},{1,1,1,1,1,1,1,2},40] (* Harvey P. Dale, Jul 29 2024 *)

a(n) = sum(k=0, n\7, binomial(n+k, n-7*k));

a(n) = A017857(8*n).
a(n) = Sum_{k=0..floor(n/7)} binomial(n+k,n-7*k).
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 9*a(n-7) - a(n-8).
G.f.: 1/(1 - x - x^7/(1 - x)^7).

cp's OEIS Frontend

A373912 Number of compositions of 7*n into parts 6 and 7.

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A107025 Binomial transform of the expansion of 1/(1-x^5-x^6).

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

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A369836 Number of compositions of 5*n into parts 1 and 5.

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

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A369849 Number of compositions of 5*n-1 into parts 4 and 5.

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A369850 Number of compositions of 5*n-2 into parts 4 and 5.

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A369851 Number of compositions of 5*n-3 into parts 4 and 5.

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A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2k,n-4k).