A017827 -id:A017827 - cp's OEIS frontend

A099131 Quintisection and binomial transform of 1/(1-x^4-x^5).

1, 1, 1, 1, 2, 7, 22, 57, 128, 264, 529, 1079, 2290, 5022, 11148, 24633, 53824, 116472, 250880, 540536, 1167937, 2531061, 5494247, 11928731, 25880583, 56101768, 121544393, 263289438, 570427339, 1236159756, 2679343966, 5807782301

Offset: 0

Paul Barry, Sep 29 2004

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-4,1).

LinearRecurrence[{5, -10, 10, -4, 1}, {1, 1, 1, 1, 2}, 32] (* Jean-François Alcover, Sep 21 2017 *)

G.f.: (1-x)^4/((1-x)^5-x^4); a(n)=sum{k=0..floor(5n/4), binomial(k, 5n-4k)}; a(n)=A017827(5n).

a(n)=sum{k=0..floor((n+1)/2), binomial(n+k, 5k)}; - Paul Barry, May 09 2005

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).

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

A017867 Expansion of 1/(1 - x^8 - x^9).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 2, 9

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts 8 and 9. - Joerg Arndt, Jun 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1,1).

Column k=8 of A306713.

Cf. A017817, A017827, A017857, A017877.

m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^8-x^9))); // Vincenzo Librandi, Jun 28 2013

I:=[1,0,0,0,0,0,0,0,1]; [n le 9 select I[n] else Self(n-8)+Self(n-9): n in [1..80]]; // Vincenzo Librandi, Jun 28 2013

CoefficientList[Series[1 / (1 - Total[x^Range[8, 9]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jun 28 2013 *)

x='x+O('x^66); Vec(1/(1-x^8-x^9)) \\ Altug Alkan, Oct 07 2018

a(n) = a(n-8) + a(n-9) for n>8. - Vincenzo Librandi, Jun 28 2013

a(n) = Sum_{k=0..floor(n/8)} binomial(k,n-8*k). - Seiichi Manyama, Oct 01 2024

A124789 Expansion of (1+x^2)/(1-x^4-x^5).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 17, 21, 27, 31, 33, 38, 48, 58, 64, 71, 86, 106, 122, 135, 157, 192, 228, 257, 292, 349, 420, 485, 549, 641, 769, 905, 1034, 1190, 1410, 1674, 1939, 2224, 2600, 3084, 3613

Offset: 0

Paul Barry, Nov 07 2006

Diagonal sums of A124788.

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

Cf. A017827, A103372.

CoefficientList[Series[(1+x^2)/(1-x^4-x^5),{x,0,60}],x] (* or *) LinearRecurrence[ {0,0,0,1,1},{1,0,1,0,1},60] (* Harvey P. Dale, Aug 20 2013 *)

a(n) = Sum_{k=0..floor(n/2)} C(floor(k/2),n-2*k).
a(n) = A017827(n)+A017827(n-2). - R. J. Mathar, May 09 2013
a(n) = A103372(n-3) for n >= 4. - Georg Fischer, Nov 03 2018
a(n) = (-1)^n*A124746(n). - R. J. Mathar, Jun 30 2020

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).

A339087 Number of compositions (ordered partitions) of n into distinct parts congruent to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 4, 1, 0, 0, 6, 4, 1, 0, 0, 6, 6, 1, 0, 0, 12, 6, 1, 0, 0, 18, 8, 1, 0, 24, 24, 8, 1, 0, 24, 30, 10, 1, 0, 48, 42, 10, 1, 0, 72, 48, 12, 1, 0, 120, 60, 12, 1, 120, 144, 72, 14, 1, 120, 216, 84, 14, 1, 240

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(27) = 6 because we have [14, 9, 4], [14, 4, 9], [9, 14, 4], [9, 4, 14], [4, 14, 9] and [4, 9, 14].

Index entries for sequences related to compositions

Cf. A016897, A017827, A032020, A032021, A109700, A281243, A337547, A337548, A339059, A339060, A339086, A339088, A339089.

nmax = 80; CoefficientList[Series[Sum[k! x^(k (5 k + 3)/2)/Product[1 - x^(5 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(5*k + 3)/2) / Product_{j=1..k} (1 - x^(5*j)).

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).

A376546 G.f. A(x) satisfies A(x) = 1 + x^4(1+x)A(x)^4.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 4, 8, 4, 0, 22, 66, 66, 22, 140, 560, 840, 560, 1109, 4845, 9690, 9690, 11929, 43473, 106260, 141680, 160080, 419244, 1137304, 1883700, 2304432, 4496076, 12157236, 23614812, 32813500, 53821332, 132821856, 285795696, 451409380

Offset: 0

Seiichi Manyama, Sep 27 2024

nonn

Cf. A017827, A366589, A375691, A376487.

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

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(4*k,k)/(3*k+1).

cp's OEIS Frontend

A099131 Quintisection and binomial transform of 1/(1-x^4-x^5).

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

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A017867 Expansion of 1/(1 - x^8 - x^9).

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A124789 Expansion of (1+x^2)/(1-x^4-x^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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A339087 Number of compositions (ordered partitions) of n into distinct parts congruent to 4 mod 5.

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A376546 G.f. A(x) satisfies A(x) = 1 + x^4(1+x)A(x)^4.