A052920 -id:A052920 - cp's OEIS frontend

A369804 Expansion of 1/(1 - x^3/(1-x)^5).

1, 0, 0, 1, 5, 15, 36, 80, 181, 431, 1060, 2617, 6401, 15521, 37513, 90741, 219918, 533619, 1295022, 3141826, 7619870, 18478155, 44810670, 108676262, 263576791, 639267800, 1550434777, 3760269946, 9119740067, 22118021213, 53642768716, 130099857234, 315531401964

Offset: 0

Seiichi Manyama, Feb 01 2024

Number of compositions of 5*n-3 into parts 3 and 5.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-5,1).

Cf. A079675, A368475, A369803.
Cf. A369845, A369846, A369847, A369848.
Cf. A052920.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^3/(1-x)^5))

a(n) = sum(k=0, n\3, binomial(n-1+2*k, n-3*k));

a(n) = A052920(5*n-3) for n > 0.
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/3)} binomial(n-1+2*k,n-3*k).
a(n) = A369845(n) - A369845(n-1). - R. J. Mathar, Feb 14 2024

A369845 Number of compositions of 5*n into parts 3 and 5.

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 58, 138, 319, 750, 1810, 4427, 10828, 26349, 63862, 154603, 374521, 908140, 2203162, 5344988, 12964858, 31443013, 76253683, 184929945, 448506736, 1087774536, 2638209313, 6398479259, 15518219326, 37636240539, 91279009255, 221378866489

Offset: 0

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-5,1).

Cf. A369804, A369846, A369847, A369848.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {1, 1, 1, 2, 7}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+2*k, n-3*k));

a(n) = A052920(5*n).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,n-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x^3).
a(n) = A369847(n+1)-A369847(n). - R. J. Mathar, Feb 14 2024

A369846 Number of compositions of 5*n-1 into parts 3 and 5.

Original entry on oeis.org

0, 1, 4, 10, 21, 44, 101, 250, 629, 1557, 3784, 9120, 21992, 53228, 129177, 313701, 761403, 1846804, 4478044, 10858285, 26332515, 63865592, 154900529, 375691009, 911166977, 2209835169, 5359470121, 12998281146, 31524747503, 76457088518, 185431544730

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,-10,11,-5,1).

Cf. A369804, A369845, A369847, A369848.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {0, 1, 4, 10, 21}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+1+2*k, n-2-3*k));

a(n) = A052920(5*n-1).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+1+2*k,n-2-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^2*(1-x)/((1-x)^5 - x^3).

A369847 Number of compositions of 5*n-2 into parts 3 and 5.

Original entry on oeis.org

1, 2, 3, 5, 12, 34, 92, 230, 549, 1299, 3109, 7536, 18364, 44713, 108575, 263178, 637699, 1545839, 3749001, 9093989, 22058847, 53501860, 129755543, 314685488, 763192224, 1850966760, 4489176073, 10887655332, 26405874658, 64042115197, 155321124452

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,-10,11,-5,1).

Cf. A369804, A369845, A369846, A369848.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {1, 2, 3, 5, 12}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+2*k, n-1-3*k));

a(n) = A052920(5*n-2).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,n-1-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x^3).

A369848 Number of compositions of 5*n-4 into parts 3 and 5.

Original entry on oeis.org

0, 1, 3, 6, 11, 23, 57, 149, 379, 928, 2227, 5336, 12872, 31236, 75949, 184524, 447702, 1085401, 2631240, 6380241, 15474230, 37533077, 91034937, 220790480, 535475968, 1298668192, 3149634952, 7638811025, 18526466357, 44932341015, 108974456212, 264295580664

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,-10,11,-5,1).

Cf. A369804, A369845, A369846, A369847.

Cf. A052920.

LinearRecurrence[{5, -10, 11, -5, 1}, {0, 1, 3, 6, 11}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\3, binomial(n+2*k, n-2-3*k));

a(n) = A052920(5*n-4).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,n-2-3*k).
a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^2*(1-x)^2/((1-x)^5 - x^3).

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A243254 Number of compositions of n into parts {3,4,5} when all parts 3,4 and 5 are present.

Original entry on oeis.org

6, 0, 0, 12, 12, 12, 20, 30, 50, 60, 80, 120, 162, 225, 305, 401, 560, 763, 1017, 1365, 1834, 2484, 3328, 4420, 5936, 7943, 10593, 14148, 18828, 25092, 33468, 44517, 59214, 78734, 104698, 139232, 184889, 245532, 326177, 433052, 574841, 762856, 1012219, 1343160

Offset: 12

David Neil McGrath, Jul 30 2014

Compositions of n from the set {3,4,5} that can be partitioned into the equivalence classes [345][34][45][35][3][4][5], where each class is defined by the relation "all elements are present".

			a(24) = 162 = 42 + 90 + 30: the tuples are (5433333) -> 7!/5! = 42, (554433) -> 6!/2!2!2! = 90, (544443) -> 6!/4! = 30.

Robert Israel, Table of n, a(n) for n = 12..7366
Index entries for linear recurrences with constant coefficients, signature (-2, -2, 2, 9, 16, 14, -2, -29, -52, -52, -20, 34, 82, 97, 67, 7, -53, -84, -77, -43, -4, 22, 29, 23, 13, 5, 1).

Cf. A245492, A245487, A245527, A022003, A011765, A112765, A017818.

N:= 100;
C34:= Vector(N):
C35:= Vector(N):
C45:= Vector(N):
C345:= Vector(N):
C1:= Vector(N,i -> numboccur([i mod 3, i mod 4, i mod 5],0)):
C34[3]:= 1: C34[4]:= 1:
C35[3]:= 1: C35[5]:= 1:
C45[4]:= 1: C45[5]:= 1:
C345[3]:= 1: C345[4]:= 1: C345[5]:= 1:
for n from 6 to N do
  C34[n]:= C34[n-3] + C34[n-4];
  C35[n]:= C35[n-3] + C35[n-5];
  C45[n]:= C45[n-4] + C45[n-5];
  C345[n]:= C345[n-3]+C345[n-4]+C345[n-5];
od:
A:= C345 - C34 - C35 - C45 + C1:
convert(A[12..N],list); # Robert Israel, Aug 18 2014

CoefficientList[Series[x^12*(x^15 + 5*x^14 + 13*x^13 + 24*x^12 + 34*x^11 + 36*x^10 + 24*x^9 - 26*x^7 - 40*x^6 - 36*x^5 - 18*x^4 + 12*x^2 + 12*x +6)/((1 - x)*(x + 1)*(x^2 + 1)*(x^3 + x^2 - 1)*(x^4 + x^3 - 1)*(x^5 + x^3 - 1)*(x^2 + x + 1)*(x^5 + x^4 - 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Aug 02 2014 *)
Drop[LinearRecurrence[{-2,-2,2,9,16,14,-2,-29,-52,-52,-20,34,82,97,67,7,-53,-84,-77,-43,-4,22,29,23,13,5,1},{0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,12,12,20,30,50,60,80,120,162,225,305,401},60],12] (* Harvey P. Dale, Jun 06 2025 *)

a(n) = A017818(n-1) -A245492(n) -A245487(n) -A245527(n) -A022003(n) -A011765(n) -A112765(n).
G.f.: -(x^15 +5*x^14 +13*x^13 +24*x^12 +34*x^11 +36*x^10 +24*x^9-26*x^7 -40*x^6 -36*x^5 -18*x^4 +12*x^2 +12*x +6) *x^12 /((x-1) *(x+1) *(x^2+1) *(x^3+x^2-1) *(x^4+x^3-1) *(x^5+x^3-1) *(x^2+x+1) *(x^5+x^4-1) *(x^4+x^3+x^2+x+1)). - Alois P. Heinz, Jul 30 2014
a(n) = A017818(n) - A017817(n) - A052920(n) - A017827(n) + A079978(n) + A121262(n) + A079998(n). - Robert Israel, Aug 18 2014

A376784 Expansion of 1/sqrt((1 - x^3 - x^5)^2 - 4*x^8).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 4, 1, 1, 9, 1, 9, 16, 2, 36, 25, 17, 100, 37, 101, 225, 74, 401, 442, 289, 1226, 820, 1306, 3138, 1737, 5000, 7106, 5161, 15998, 15185, 18901, 44308, 34282, 67861, 110365, 92501, 219609, 259621, 295242, 637570, 620350, 986373, 1694619

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Cf. A246884, A376722, A376783.

Cf. A052920.

CoefficientList[Series[1/Sqrt[(1-x^3-x^5)^2-4x^8],{x,0,50}],x] (* Harvey P. Dale, Apr 28 2025 *)

my(N=50, x='x+O('x^N)); Vec(1/sqrt((1-x^3-x^5)^2-4*x^8))

a(n) = sum(k=0, n\5, ((n-2*k)%3==0)*binomial((n-2*k)/3, k)^2);

G.f.: 1/sqrt((1 - x^3 + x^5)^2 - 4*x^5) = 1/sqrt((1 + x^3 - x^5)^2 - 4*x^3).

cp's OEIS Frontend

A369804 Expansion of 1/(1 - x^3/(1-x)^5).

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A369845 Number of compositions of 5*n into parts 3 and 5.

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A369846 Number of compositions of 5*n-1 into parts 3 and 5.

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A369847 Number of compositions of 5*n-2 into parts 3 and 5.

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A369848 Number of compositions of 5*n-4 into parts 3 and 5.

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A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A243254 Number of compositions of n into parts {3,4,5} when all parts 3,4 and 5 are present.

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A376784 Expansion of 1/sqrt((1 - x^3 - x^5)^2 - 4*x^8).

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