A099253 -id:A099253 - cp's OEIS frontend

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

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

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).

A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 21, 13, 5, 1, 1, 32, 55, 41, 19, 6, 1, 1, 64, 144, 129, 69, 26, 7, 1, 1, 128, 377, 406, 250, 106, 34, 8, 1, 1, 256, 987, 1278, 907, 431, 153, 43, 9, 1, 1, 512, 2584, 4023, 3292, 1757, 686, 211, 53, 10, 1, 1, 1024, 6765, 12664, 11949, 7168, 3088, 1030, 281, 64, 11, 1

Offset: 0

Paul Barry, Oct 08 2004

Rows include A099242, A099253. Columns include A034856. Main diagonal is A099240. Sums of antidiagonals are A099241.

			Rows begin
  1, 1,  1,   1,   1, ...                               A000012;
  1, 2,  4,   8,  16, ...      1-section of 1/(1-x-x)   A000079;
  1, 3,  8,  21,  55, ....     bisection of 1/(1-x-x^2) A001906;
  1, 4, 13,  41, 129, ...     trisection of 1/(1-x-x^3) A052529; (essentially)
  1, 5, 19,  69, 250, ...  quadrisection of 1/(1-x-x^4) A055991;
  1, 6, 26, 106, 431, ...  quintisection of 1/(1-x-x^5) A079675; (essentially)

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Cf. A034856, A099240, A099241, A099242, A099253.

Cf. A000079, A001906, A052529, A055991, A079675.

A099239:= func< n,k | (&+[Binomial(k*(n-k) -(k-1)*(j-1), j): j in [0..n-k]]) >;
[A099239(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

T[n_, k_]:= Sum[Binomial[k*(n-k) - (k-1)*(j-1), j], {j,0,n-k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

def A099239(n,k): return sum( binomial(k*(n-k) -(k-1)*(j-1), j) for j in (0..n-k) )
flatten([[A099239(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

T(n, k) = Sum_{j=0..n} binomial(k*n -(k-1)*(j-1), j), n, k>=0. (square array)
T(n, k) = Sum_{j=0..n} binomial(k + (n-1)*(j+1), n*(j+1) -1), n>0. (square array)
T(n, k) = Sum_{j=0..n-k} binomial(k*(n-k) - (k-1)*(j-1), j). (number triangle)
Rows of the square array are generated by 1/((1-x)^k-x).
Rows satisfy a(n) = a(n-1) - Sum_{k=1..n} (-1)^(k^binomial(n, k)) * a(n-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).

A373928 Number of compositions of 7*n-2 into parts 1 and 7.

Original entry on oeis.org

1, 7, 35, 168, 819, 4025, 19796, 97315, 478304, 2350860, 11554621, 56791883, 279136551, 1371977475, 6743373646, 33144194898, 162906243014, 800696596250, 3935484773527, 19343207491818, 95073338508548, 467292702057555, 2296779231936167, 11288844908179562

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373929, A373930, A373931, A373932.

Cf. A005709, A369805, A369837.

a[n_]:= n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*HypergeometricPFQ[{1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6}, {6/7, 8/7, 9/7, 10/7, 11/7, 12/7}, -6^6/7^7]/120; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
LinearRecurrence[{8,-21,35,-35,21,-7,1},{1,7,35,168,819,4025,19796},40] (* Harvey P. Dale, Jul 28 2024 *)

a(n) = sum(k=0, n, binomial(n+4+6*k, n-1-k));

a(n) = A005709(7*n-2).
a(n) = Sum_{k=0..n} binomial(n+4+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6], [6/7, 8/7, 9/7, 10/7, 11/7, 12/7], -6^6/7^7)/120. - Stefano Spezia, Jun 23 2024

A373929 Number of compositions of 7*n-3 into parts 1 and 7.

Original entry on oeis.org

1, 6, 28, 133, 651, 3206, 15771, 77519, 380989, 1872556, 9203761, 45237262, 222344668, 1092840924, 5371396171, 26400821252, 129762048116, 637790353236, 3134788177277, 15407722718291, 75730131016730, 372219363549007, 1829486529878612, 8992065676243395

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373928, A373930, A373931, A373932.

Cf. A005709, A369806.

a[n_]:=n*(1 + n)*(2 + n)*(3 + n)*HypergeometricPFQ[{1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6}, {5/7, 6/7, 8/7, 9/7, 10/7, 11/7}, -6^6/7^7]/24; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)

a(n) = sum(k=0, n, binomial(n+3+6*k, n-1-k));

a(n) = A005709(7*n-3).
a(n) = Sum_{k=0..n} binomial(n+3+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^2/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*(3 + n)*hypergeom([1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6], [5/7, 6/7, 8/7, 9/7, 10/7, 11/7], -6^6/7^7)/24. - Stefano Spezia, Jun 23 2024

A373930 Number of compositions of 7*n-4 into parts 1 and 7.

Original entry on oeis.org

1, 5, 22, 105, 518, 2555, 12565, 61748, 303470, 1491567, 7331205, 36033501, 177107406, 870496256, 4278555247, 21029425081, 103361226864, 508028305120, 2496997824041, 12272934541014, 60322408298439, 296489232532277, 1457267166329605, 7162579146364783

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373928, A373929, A373931, A373932.

Cf. A005709, A369807.

a[n_]:=n*(1 + n)*(2 + n)*HypergeometricPFQ[{1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6}, {4/7, 5/7, 6/7, 8/7, 9/7, 10/7}, -6^6/7^7]/6; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)

a(n) = sum(k=0, n, binomial(n+2+6*k, n-1-k));

a(n) = A005709(7*n-4).
a(n) = Sum_{k=0..n} binomial(n+2+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^3/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*hypergeom([1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6], [4/7, 5/7, 6/7, 8/7, 9/7, 10/7], -6^6/7^7)/6. - Stefano Spezia, Jun 23 2024

A373931 Number of compositions of 7*n-5 into parts 1 and 7.

Original entry on oeis.org

1, 4, 17, 83, 413, 2037, 10010, 49183, 241722, 1188097, 5839638, 28702296, 141073905, 693388850, 3408058991, 16750869834, 82331801783, 404667078256, 1988969518921, 9775936716973, 48049473757425, 236166824233838, 1160777933797328, 5705311980035178

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Cf. A099253, A373907, A373928, A373929, A373930, A373932.

Cf. A005709, A369808.

a[n_]:=n*(1 + n)*HypergeometricPFQ[{1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6}, {3/7, 4/7, 5/7, 6/7, 8/7, 9/7}, -6^6/7^7]/2; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)

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

a(n) = A005709(7*n-5).
a(n) = Sum_{k=0..n} binomial(n+1+6*k,n-1-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1-x)^4/((1-x)^7 - x).
a(n) = n*(1 + n)*hypergeom([1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6], [3/7, 4/7, 5/7, 6/7, 8/7, 9/7], -6^6/7^7)/2. - Stefano Spezia, Jun 23 2024

cp's OEIS Frontend

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

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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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A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k).

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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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A373928 Number of compositions of 7*n-2 into parts 1 and 7.

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A373929 Number of compositions of 7*n-3 into parts 1 and 7.