A099132 -id:A099132 - cp's OEIS frontend

A017837 Expansion of 1/(1 - x^5 - x^6).

1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 2, 6, 15, 20, 15, 7, 8, 21, 35, 35, 22, 15, 29, 56, 70, 57, 37, 44, 85, 126, 127, 94, 81, 129, 211, 253, 221, 175, 210, 340, 464, 474, 396, 385, 550, 804, 938, 870, 781, 935, 1354, 1742, 1808, 1651, 1716, 2289, 3096

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts 5 and 6. - Joerg Arndt, Jun 27 2013

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,1).

Column 5 of A306713.

Cf. A099132.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^5-x^6))); // Vincenzo Librandi, Jun 27 2013

I:=[1,0,0,0,0,1]; [n le 6 select I[n] else Self(n-5)+Self(n-6): n in [1..70]]; // Vincenzo Librandi, Jun 27 2013

CoefficientList[Series[1 / (1 - Total[x^Range[5, 6]]), {x, 0, 50}], x] (* Vincenzo Librandi Jun 27 2013 *)

Vec(1/(1-x^5-x^6)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = a(n-6) + a(n-5). - Jon E. Schoenfield, Aug 07 2006

A306680 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^(k+1)).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 5, 1, 1, 1, 2, 5, 6, 1, 1, 1, 1, 4, 8, 7, 1, 1, 1, 1, 2, 7, 13, 8, 1, 1, 1, 1, 1, 5, 12, 21, 9, 1, 1, 1, 1, 1, 2, 11, 21, 34, 10, 1, 1, 1, 1, 1, 1, 6, 21, 37, 55, 11, 1, 1, 1, 1, 1, 1, 2, 16, 37, 65, 89, 12

Offset: 0

Seiichi Manyama, Mar 05 2019

			A(4,1) = A306713(4,1) = 5, A(4,2) = A306713(8,2) = 4.
Square array begins:
   1,  1,  1,  1,  1,  1, 1, 1, 1, ...
   2,  1,  1,  1,  1,  1, 1, 1, 1, ...
   3,  2,  1,  1,  1,  1, 1, 1, 1, ...
   4,  3,  2,  1,  1,  1, 1, 1, 1, ...
   5,  5,  4,  2,  1,  1, 1, 1, 1, ...
   6,  8,  7,  5,  2,  1, 1, 1, 1, ...
   7, 13, 12, 11,  6,  2, 1, 1, 1, ...
   8, 21, 21, 21, 16,  7, 2, 1, 1, ...
   9, 34, 37, 37, 36, 22, 8, 2, 1, ...

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

Columns 0-9 give A000027(n+1), A000045(n+1), A005251(n+1), A003522, A005676, A099132, A293169, A306721, A306752, A306753.

Cf. A306646, A306713, A306735.

T[n_, k_] := Sum[Binomial[n - j, k*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

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

A(n,k) = A306713(k*n,k) for k > 0.

A293169 a(n) = Sum_{k=0..n} binomial(k, 6*(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 925, 1718, 3017, 5097, 8464, 14197, 24753, 45697, 89150, 180254, 368734, 748924, 1493990, 2914906, 5565127, 10434412, 19322901, 35583926, 65615746, 121847272, 228638698, 433747259, 830227401, 1597653852, 3078928619, 5922703731, 11347651254

Offset: 0

N. J. A. Sloane, Oct 17 2017

Colin Barker, Table of n, a(n) for n = 0..1000
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1).

Cf. A005676, A099132.

f:=n-> add( binomial(k, 6*(n-k)), k=0..n);
[seq(f(n),n=0..30)];

Table[Sum[Binomial[k,6(n-k)],{k,0,n}],{n,0,40}] (* or *)  LinearRecurrence[{6,-15,20,-15,6,-1,1},{1,1,1,1,1,1,1},50] (* Harvey P. Dale, Apr 10 2022 *)

Vec((1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7) + O(x^30)) \\ Colin Barker, Oct 18 2017

From Colin Barker, Oct 17 2017: (Start)
G.f.: (1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + a(n-7) for n>6.
(End)

cp's OEIS Frontend

A017837 Expansion of 1/(1 - x^5 - x^6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

A306680 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^(k+1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A293169 a(n) = Sum_{k=0..n} binomial(k, 6*(n-k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula