A099099 -id:A099099 - 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

A373905 a(n) = Sum_{k=0..floor(n/3)} binomial(n+3k,n-3k).

Original entry on oeis.org

1, 1, 1, 2, 8, 29, 86, 224, 554, 1381, 3556, 9382, 24901, 65737, 172321, 450017, 1174985, 3072365, 8044478, 21074012, 55199573, 144535714, 378366976, 990441502, 2592800365, 6787973872, 17771619370, 46527959417, 121813193825, 318910531073, 834913179137

Offset: 0

Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,21,-15,6,-1).

Cf. A000930, A005251, A024493, A099099, A369845.

Cf. A107025, A373904, A373906.

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

a(n) = 6*a(n-1) - 15*a(n-2) + 21*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

G.f.: 1/(1 - x - x^3/(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).

A099101 Quintisection of 1/(1-x^3-x^4).

Original entry on oeis.org

1, 0, 3, 5, 16, 43, 113, 316, 839, 2301, 6204, 16855, 45665, 123800, 335659, 909845, 2466760, 6686979, 18128529, 49145300, 133231279, 361184653, 979156724, 2654456239, 7196122817, 19508406192, 52886508243, 143373224101

Offset: 0

Paul Barry, Sep 29 2004

Index entries for linear recurrences with constant coefficients, signature (0,5,6,1).

Cf. A099099.

a(n)=sum{k=0..floor(5n/3), binomial(k, 5n-3k)}.
a(n)=A017817(5n).
G.f.: (1+x)*(x^2+x-1) / ( -1+5*x^2+6*x^3+x^4 ). - R. J. Mathar, Feb 19 2015

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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A373905 a(n) = Sum_{k=0..floor(n/3)} binomial(n+3*k,n-3*k).

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

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A099101 Quintisection of 1/(1-x^3-x^4).

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A373905 a(n) = Sum_{k=0..floor(n/3)} binomial(n+3k,n-3k).