A079675 -id:A079675 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A369836 Number of compositions of 5*n into parts 1 and 5.

Original entry on oeis.org

1, 2, 8, 34, 140, 571, 2328, 9496, 38740, 158045, 644761, 2630364, 10730820, 43777405, 178594110, 728591751, 2972359720, 12126025705, 49469281395, 201814663875, 823322219501, 3358821723401, 13702634402876, 55901207340276, 228054320813276, 930369409108152

Offset: 0

Seiichi Manyama, Feb 03 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,10,-5,1).

Cf. A079675, A369837, A369838, A369839.
Cf. A099131, A369840, A369845.
Cf. A003520.

LinearRecurrence[{6, -10, 10, -5, 1}, {1, 2, 8, 34, 140}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A003520(5*n).
a(n) = Sum_{k=0..n} binomial(n+4*k,n-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x).

A369803 Expansion of 1/(1 - x^2/(1-x)^5).

Original entry on oeis.org

1, 0, 1, 5, 16, 45, 126, 361, 1046, 3032, 8771, 25348, 73252, 211724, 612009, 1769080, 5113647, 14781237, 42725841, 123501151, 356986401, 1031887518, 2982723523, 8621714049, 24921502864, 72036871920, 208226244217, 601888555723, 1739789499591, 5028950081882

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,10,-5,1).

Cf. A079675, A368475, A369804.
Cf. A369840, A369842, A369843, A369844.
Cf. A001687.

my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^5))

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

a(n) = A001687(5*n-1) for n > 0.
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+3*k,n-2*k).
a(n) = A369840(n)-A369840(n-1). - R. J. Mathar, Feb 14 2024

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).

A369837 Number of compositions of 5*n-2 into parts 1 and 5.

Original entry on oeis.org

1, 5, 20, 80, 325, 1326, 5411, 22076, 90061, 367411, 1498887, 6114853, 24946129, 101770120, 415180936, 1693770328, 6909898016, 28189589705, 115002126790, 469162173146, 1913991948274, 7808313175575, 31854760257925, 129954540535600, 530161974821876

Offset: 1

Seiichi Manyama, Feb 03 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,10,-5,1).

Cf. A079675, A369836, A369838, A369839.

Cf. A003520, A055990.

LinearRecurrence[{6, -10, 10, -5, 1}, {1, 5, 20, 80, 325}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n, binomial(n+2+4*k, n-1-k));

a(n) = A003520(5*n-2).
a(n) = Sum_{k=0..n} binomial(n+2+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)/((1-x)^5 - x).

A369838 Number of compositions of 5*n-3 into parts 1 and 5.

Original entry on oeis.org

1, 4, 15, 60, 245, 1001, 4085, 16665, 67985, 277350, 1131476, 4615966, 18831276, 76823991, 313410816, 1278589392, 5216127688, 21279691689, 86812537085, 354160046356, 1444829775128, 5894321227301, 24046447082350, 98099780277675, 400207434286276, 1632684497403029

Offset: 1

Seiichi Manyama, Feb 03 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,10,-5,1).

Cf. A079675, A369836, A369837, A369839.

Cf. A003520.

LinearRecurrence[{6, -10, 10, -5, 1}, {1, 4, 15, 60, 245}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A003520(5*n-3).
a(n) = Sum_{k=0..n} binomial(n+1+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^2/((1-x)^5 - x).

A369839 Number of compositions of 5*n-4 into parts 1 and 5.

Original entry on oeis.org

1, 3, 11, 45, 185, 756, 3084, 12580, 51320, 209365, 854126, 3484490, 14215310, 57992715, 236586825, 965178576, 3937538296, 16063564001, 65532845396, 267347509271, 1090669728772, 4449491452173, 18152125855049, 74053333195325, 302107654008601, 1232477063116753

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 (6,-10,10,-5,1).

Cf. A079675, A369836, A369837, A369838.

Cf. A003520.

LinearRecurrence[{6, -10, 10, -5, 1}, {1, 3, 11, 45, 185}, 50] (* Paolo Xausa, Mar 15 2024 *)

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

a(n) = A003520(5*n-4).
a(n) = Sum_{k=0..n} binomial(n+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x).

A365084 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x)^5.

Original entry on oeis.org

1, 1, -4, 6, 6, -49, 95, 24, -592, 1417, -414, -6809, 20142, -14831, -73353, 274761, -311105, -715647, 3607624, -5463428, -5785294, 45588556, -87189477, -25565196, 552659892, -1305250324, 340413165, 6379267117, -18606431142, 13202513476, 69064770845

Offset: 0

Seiichi Manyama, Aug 21 2023

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

Cf. A106510, A127896, A365083.

Cf. A079675.

LinearRecurrence[{-4, -10, -10, -5, -1}, {1, 1, -4, 6, 6, -49}, 1 + 30] (* Robert P. P. McKone, Aug 21 2023 *)

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A369836 Number of compositions of 5*n into parts 1 and 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369803 Expansion of 1/(1 - x^2/(1-x)^5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A369837 Number of compositions of 5*n-2 into parts 1 and 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369838 Number of compositions of 5*n-3 into parts 1 and 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369839 Number of compositions of 5*n-4 into parts 1 and 5.

Views