A369809 -id:A369809 - 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

A369807 Expansion of 1/(1 - x^4/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 0, 1, 7, 28, 84, 211, 476, 1029, 2276, 5384, 13594, 35371, 91667, 232681, 577710, 1413462, 3442498, 8414484, 20717963, 51346109, 127678961, 317496621, 787941379, 1950774874, 4821609252, 11910608942, 29432604429, 72787392898, 180131835001

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Number of compositions of 7*n-4 into parts 4 and 7.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-34,21,-7,1).

Cf. A099253, A369805, A369806, A369808, A369809.

Cf. A369815.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^4/(1-x)^7))

a(n) = sum(k=0, n\4, binomial(n-1+3*k, n-4*k));

a(n) = A369815(7*n-4) for n > 0.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 34*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-1+3*k,n-4*k).

A369808 Expansion of 1/(1 - x^5/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 7, 28, 84, 210, 463, 938, 1821, 3563, 7385, 16577, 39529, 96315, 232393, 546806, 1251461, 2801015, 6189683, 13647361, 30281870, 67918782, 153939843, 351309676, 803438125, 1834160110, 4170751775, 9443922772, 21316094357, 48041401423, 108291578580

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Number of compositions of 7*n-5 into parts 5 and 7.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,22,-7,1).

Cf. A099253, A369805, A369806, A369807, A369809.

Cf. A369816.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^5/(1-x)^7))

a(n) = sum(k=0, n\5, binomial(n-1+2*k, n-5*k));

a(n) = A369816(7*n-5) for n > 0.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 22*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/5)} binomial(n-1+2*k,n-5*k).

A369805 Expansion of 1/(1 - x^2/(1-x)^7).

Original entry on oeis.org

1, 0, 1, 7, 29, 98, 316, 1043, 3536, 12083, 41168, 139750, 473824, 1607014, 5453022, 18506947, 62808496, 213144034, 723295969, 2454483506, 8329290739, 28265565587, 95919580313, 325504019213, 1104600373788, 3748469764612, 12720462563684, 43166996581876

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Number of compositions of 7*n-2 into parts 2 and 7.

Index entries for linear recurrences with constant coefficients, signature (7,-20,35,-35,21,-7,1).

Cf. A099253, A369806, A369807, A369808, A369809.

Cf. A369813.

my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^7))

a(n) = sum(k=0, n\2, binomial(n-1+5*k, n-2*k));

a(n) = A369813(7*n-2) for n > 0.
a(n) = 7*a(n-1) - 20*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+5*k,n-2*k).

A369806 Expansion of 1/(1 - x^3/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 1, 7, 28, 85, 224, 567, 1485, 4117, 11802, 33909, 96182, 269402, 750275, 2090728, 5845015, 16384908, 45973701, 128944042, 361364501, 1012168575, 2834690172, 7939970075, 22244001961, 62323608147, 174620915138, 489240430938, 1370662332271, 3839992876850

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Number of compositions of 7*n-3 into parts 3 and 7.

Index entries for linear recurrences with constant coefficients, signature (7,-21,36,-35,21,-7,1).

Cf. A099253, A369805, A369807, A369808, A369809.

Cf. A369814.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^3/(1-x)^7))

a(n) = sum(k=0, n\3, binomial(n-1+4*k, n-3*k));

a(n) = A369814(7*n-3) for n > 0.
a(n) = 7*a(n-1) - 21*a(n-2) + 36*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/3)} binomial(n-1+4*k,n-3*k).

A373933 Number of compositions of 7*n-1 into parts 6 and 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 17, 54, 175, 506, 1299, 3017, 6465, 13021, 25142, 47651, 91104, 180254, 374077, 810381, 1800140, 4019204, 8888489, 19322901, 41223071, 86520282, 179574728, 370946309, 767426451, 1597653852, 3354537225, 7101005320, 15118658953

Offset: 1

Seiichi Manyama, Jun 23 2024

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

Cf. A369809, A373912, A373934, A373935, A373936, A373937.

Cf. A017847, A369849.

a(n) = sum(k=0, n\6, binomial(n+k, n-1-6*k));

a(n) = A017847(7*n-1).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-1-6*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.: x*(1-x)^5/((1-x)^7 - x^6).
a(n) = A373934(n+1)-A373934(n). - R. J. Mathar, Jun 24 2024

A373934 Number of compositions of 7*n-2 into parts 6 and 7.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 29, 46, 100, 275, 781, 2080, 5097, 11562, 24583, 49725, 97376, 188480, 368734, 742811, 1553192, 3353332, 7372536, 16261025, 35583926, 76806997, 163327279, 342902007, 713848316, 1481274767, 3078928619, 6433465844, 13534471164

Offset: 1

Seiichi Manyama, Jun 23 2024

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

Cf. A369809, A373912, A373933, A373935, A373936, A373937.

Cf. A017847, A369850.

a(n) = sum(k=0, n\6, binomial(n+k, n-2-6*k));

a(n) = A017847(7*n-2).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-2-6*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.: x^2*(1-x)^4/((1-x)^7 - x^6).
a(n) = A373935(n+1)-A373935(n). - R. J. Mathar, Jun 24 2024

A373935 Number of compositions of 7*n-3 into parts 6 and 7.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 56, 85, 131, 231, 506, 1287, 3367, 8464, 20026, 44609, 94334, 191710, 380190, 748924, 1491735, 3044927, 6398259, 13770795, 30031820, 65615746, 142422743, 305750022, 648652029, 1362500345, 2843775112, 5922703731, 12356169575

Offset: 1

Seiichi Manyama, Jun 23 2024

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

Cf. A369809, A373912, A373933, A373934, A373936, A373937.

Cf. A017847.

a(n) = sum(k=0, n\6, binomial(n+k, n-3-6*k));

a(n) = A017847(7*n-3).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-3-6*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.: x^3*(1-x)^3/((1-x)^7 - x^6).
a(n) = A373936(n+1)-A373936(n). - R. J. Mathar, Jun 24 2024

A373936 Number of compositions of 7*n-4 into parts 6 and 7.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 70, 126, 211, 342, 573, 1079, 2366, 5733, 14197, 34223, 78832, 173166, 364876, 745066, 1493990, 2985725, 6030652, 12428911, 26199706, 56231526, 121847272, 264270015, 570020037, 1218672066, 2581172411, 5424947523, 11347651254

Offset: 1

Seiichi Manyama, Jun 23 2024

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

Cf. A369809, A373912, A373933, A373934, A373935, A373937.

Cf. A017847, A369807.

a(n) = sum(k=0, n\6, binomial(n+k, n-4-6*k));

a(n) = A017847(7*n-4).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-4-6*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.: x^4*(1-x)^2/((1-x)^7 - x^6).
a(n) = A373937(n+1)-A373937(n). - R. J. Mathar, Jun 24 2024

A373937 Number of compositions of 7*n-5 into parts 6 and 7.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 805, 1378, 2457, 4823, 10556, 24753, 58976, 137808, 310974, 675850, 1420916, 2914906, 5900631, 11931283, 24360194, 50559900, 106791426, 228638698, 492908713, 1062928750, 2281600816, 4862773227, 10287720750

Offset: 1

Seiichi Manyama, Jun 23 2024

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

Cf. A369809, A373912, A373933, A373934, A373935, A373936.

Cf. A017847, A369808.

a(n) = sum(k=0, n\6, binomial(n+k, n-5-6*k));

a(n) = A017847(7*n-5).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-5-6*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.: x^5*(1-x)/((1-x)^7 - x^6).
a(n) = A369809(n+1)-A369809(n). - R. J. Mathar, Jun 24 2024

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

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A369807 Expansion of 1/(1 - x^4/(1-x)^7).

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A369808 Expansion of 1/(1 - x^5/(1-x)^7).

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A369805 Expansion of 1/(1 - x^2/(1-x)^7).

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A369806 Expansion of 1/(1 - x^3/(1-x)^7).

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A373933 Number of compositions of 7*n-1 into parts 6 and 7.

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A373934 Number of compositions of 7*n-2 into parts 6 and 7.

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A373935 Number of compositions of 7*n-3 into parts 6 and 7.

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