A017847 -id:A017847 - 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).

A369809 Expansion of 1/(1 - x^6/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 925, 1730, 3108, 5565, 10388, 20944, 45697, 104673, 242481, 553455, 1229305, 2650221, 5565127, 11465758, 23397041, 47757235, 98317135, 205108561, 433747259, 926655972, 1989584722, 4271185538, 9133958765, 19421679515

Offset: 0

Seiichi Manyama, Feb 01 2024

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

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

Cf. A099253, A369805, A369806, A369807, A369808.
Cf. A088305, A095263, A290998, A368475, A369794.
Cf. A000579, A017847.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^6/(1-x)^7))

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

G.f. (1-x)^7/((1-x)^7-x^6).
a(n) = A017847(7*n-6) = Sum_{k=0..floor((7*n-6)/6)} binomial(k,7*n-6-6*k) for n > 0.
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) for n > 7.
a(n) = Sum_{k=0..floor(n/6)} binomial(n-1+k,n-6*k).
a(n) = A373912(n)-A373912(n-1). - R. J. Mathar, Jun 24 2024

A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 1, 8, 1, 0, 0, 0, 1, 2, 13, 1, 0, 0, 0, 1, 0, 2, 21, 1, 0, 0, 0, 0, 1, 1, 3, 34, 1, 0, 0, 0, 0, 1, 0, 2, 4, 55, 1, 0, 0, 0, 0, 0, 1, 0, 1, 5, 89, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 7, 144, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 9, 233

Offset: 0

Seiichi Manyama, Mar 05 2019

A(n,k) is the number of compositions of n into parts k and k+1.

			Square array begins:
    1, 1, 1, 1, 1, 1, 1, 1, 1, ...
    1, 0, 0, 0, 0, 0, 0, 0, 0, ...
    2, 1, 0, 0, 0, 0, 0, 0, 0, ...
    3, 1, 1, 0, 0, 0, 0, 0, 0, ...
    5, 1, 1, 1, 0, 0, 0, 0, 0, ...
    8, 2, 0, 1, 1, 0, 0, 0, 0, ...
   13, 2, 1, 0, 1, 1, 0, 0, 0, ...
   21, 3, 2, 0, 0, 1, 1, 0, 0, ...
   34, 4, 1, 1, 0, 0, 1, 1, 0, ...
   55, 5, 1, 2, 0, 0, 0, 1, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-10 give A000045(n+1), A182097, A017817, A017827, A017837, A017847, A017857, A017867, A017877, A017887.

Cf. A306646, A306680.

T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(j,n-k*j).

A369813 Expansion of 1/(1 - x^2 - x^7).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 4, 6, 7, 7, 11, 9, 16, 13, 22, 20, 29, 31, 38, 47, 51, 69, 71, 98, 102, 136, 149, 187, 218, 258, 316, 360, 452, 509, 639, 727, 897, 1043, 1257, 1495, 1766, 2134, 2493, 3031, 3536, 4288, 5031, 6054, 7165, 8547, 10196, 12083, 14484, 17114, 20538, 24279, 29085, 34475

Offset: 0

Seiichi Manyama, Feb 02 2024

Number of compositions of n into parts 2 and 7.

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

Cf. A005709, A017847, A369814, A369815, A369816.

Cf. A007380.

CoefficientList[Series[1/(1-x^2-x^7),{x,0,80}],x] (* or *) LinearRecurrence[{0,1,0,0,0,0,1},{1,0,1,0,1,0,1},80] (* Harvey P. Dale, Dec 04 2024 *)

my(N=70, x='x+O('x^N)); Vec(1/(1-x^2-x^7))

a(n) = sum(k=0, n\7, ((n-5*k)%2==0)*binomial((n-5*k)/2, k));

a(n) = a(n-2) + a(n-7).

a(n) = A007380(n-7) for n >= 8.

A369814 Expansion of 1/(1 - x^3 - x^7).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 1, 1, 4, 3, 1, 5, 6, 2, 6, 10, 5, 7, 15, 11, 9, 21, 21, 14, 28, 36, 25, 37, 57, 46, 51, 85, 82, 76, 122, 139, 122, 173, 224, 204, 249, 346, 343, 371, 519, 567, 575, 768, 913, 918, 1139, 1432, 1485, 1714, 2200, 2398, 2632, 3339, 3830, 4117, 5053, 6030, 6515, 7685

Offset: 0

Seiichi Manyama, Feb 02 2024

Number of compositions of n into parts 3 and 7.

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

Cf. A005709, A017847, A369813, A369815, A369816.

my(N=80, x='x+O('x^N)); Vec(1/(1-x^3-x^7))

a(n) = sum(k=0, n\7, ((n-4*k)%3==0)*binomial((n-4*k)/3, k));

a(n) = a(n-3) + a(n-7).

A369815 Expansion of 1/(1 - x^4 - x^7).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 3, 1, 0, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 2, 10, 15, 7, 6, 20, 21, 9, 16, 35, 28, 15, 36, 56, 37, 31, 71, 84, 52, 67, 127, 121, 83, 138, 211, 173, 150, 265, 332, 256, 288, 476, 505, 406, 553, 808, 761, 694, 1029, 1313, 1167, 1247, 1837, 2074, 1861, 2276, 3150, 3241

Offset: 0

Seiichi Manyama, Feb 02 2024

Number of compositions of n into parts 4 and 7.

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

Cf. A005709, A017847, A369813, A369814, A369816.

my(N=80, x='x+O('x^N)); Vec(1/(1-x^4-x^7))

a(n) = sum(k=0, n\7, ((n-3*k)%4==0)*binomial((n-3*k)/4, k));

a(n) = a(n-4) + a(n-7).

A369816 Expansion of 1/(1 - x^5 - x^7).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 1, 1, 4, 0, 6, 1, 4, 5, 1, 10, 1, 10, 6, 5, 15, 2, 20, 7, 15, 21, 7, 35, 9, 35, 28, 22, 56, 16, 70, 37, 57, 84, 38, 126, 53, 127, 121, 95, 210, 91, 253, 174, 222, 331, 186, 463, 265, 475, 505, 408, 794, 451, 938, 770, 883, 1299, 859, 1732, 1221

Offset: 0

Seiichi Manyama, Feb 02 2024

Number of compositions of n into parts 5 and 7.

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

Cf. A005709, A017847, A369813, A369814, A369815.

my(N=80, x='x+O('x^N)); Vec(1/(1-x^5-x^7))

a(n) = sum(k=0, n\7, ((n-2*k)%5==0)*binomial((n-2*k)/5, k));

a(n) = a(n-5) + a(n-7).

G.f.: 1/((1-x+x^2)*(1+x-x^3-x^4-x^5)). - R. J. Mathar, Jul 03 2024

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

cp's OEIS Frontend

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

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A369809 Expansion of 1/(1 - x^6/(1-x)^7).

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A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

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A369813 Expansion of 1/(1 - x^2 - x^7).

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A369814 Expansion of 1/(1 - x^3 - x^7).

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A369815 Expansion of 1/(1 - x^4 - x^7).

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A369816 Expansion of 1/(1 - x^5 - x^7).

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

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