A306713 -id:A306713 - cp's OEIS frontend

A017847 Expansion of 1/(1 - x^6 - x^7).

1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 2, 7, 21, 35, 35, 21, 8, 9, 28, 56, 70, 56, 29, 17, 37, 84, 126, 126, 85, 46, 54, 121, 210, 252, 211, 131, 100, 175, 331, 462, 463, 342, 231, 275, 506, 793, 925

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts 6 and 7. - Joerg Arndt, Jun 27 2013

Harvey P. Dale, Table of n, a(n) for n = 0..1000
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 (0,0,0,0,0,1,1).

Column k=6 of A306713.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^6-x^7))); // Vincenzo Librandi, Jun 27 2013

I:=[1,0,0,0,0,0,1]; [n le 7 select I[n] else Self(n-6)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Jun 27 2013

CoefficientList[Series[1/(1-x^6-x^7), {x, 0, 70}], x] (* or *)  LinearRecurrence[{0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 1}, 70] (* Harvey P. Dale, Dec 15 2012 *)
CoefficientList[Series[1 / (1 - Total[x^Range[6, 7]]),{x, 0, 70}], x] (* Vincenzo Librandi, Jun 27 2013 *)

a(0)=1, a(1)=0,a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=1; for n>6, a(n) = a(n-6)+a(n-7). - Harvey P. Dale, Dec 15 2012

a(n) = Sum_{k=0..floor(n/6)} binomial(k,n-6*k). - Seiichi Manyama, Oct 01 2024

A017877 Expansion of 1/(1 - x^9 - x^10).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9 and 10. - Ilya Gutkovskiy, May 27 2017

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

Column k=9 of A306713.

Cf. A017876, A017878, A017879, A017880, A017881, A017882, A017883, A017884, A017885, A017886.

m:=80; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 10]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017877_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x^9-x^(10)) ).list()
A017877_list(85) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) + a(n-10) for n > 9. - Vincenzo Librandi, Jul 01 2013

a(n) = Sum_{k=0..floor(n/9)} binomial(k,n-9*k). - Seiichi Manyama, Oct 01 2024

A017887 Expansion of 1/(1 - x^10 - x^11).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 10 and 11. - Ilya Gutkovskiy, May 27 2017

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

Column k=10 of A306713.

Cf. A017886, A017888, A017889, A017890, A017891, A017892, A017893, A017894, A017895, A017896.

m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^10-x^11))); // Vincenzo Librandi, Jul 01 2013

I:=[1,0,0,0,0,0,0,0,0,0,1]; [n le 11 select I[n] else Self(n-10)+Self(n-11): n in [1..80]]; // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[10, 11]]), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,1,1},{1,0,0,0,0,0,0,0,0,0,1},100] (* Harvey P. Dale, Feb 04 2015 *)

def A017887_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x^10-x^(11)) ).list()
A017887_list(100) # G. C. Greubel, Sep 25 2024

a(n) = a(n-10) + a(n-11) for n > 10. - Vincenzo Librandi, Jul 01 2013

a(n) = Sum_{k=0..floor(n/10)} binomial(k,n-10*k). - Seiichi Manyama, Oct 01 2024

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

Original entry on oeis.org

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

A306646 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. (k+1-x^k)/(1-x^k-x^(k+1)).

Original entry on oeis.org

2, 3, 1, 4, 0, 3, 5, 0, 2, 4, 6, 0, 0, 3, 7, 7, 0, 0, 3, 2, 11, 8, 0, 0, 0, 4, 5, 18, 9, 0, 0, 0, 4, 0, 5, 29, 10, 0, 0, 0, 0, 5, 3, 7, 47, 11, 0, 0, 0, 0, 5, 0, 7, 10, 76, 12, 0, 0, 0, 0, 0, 6, 0, 4, 12, 123, 13, 0, 0, 0, 0, 0, 6, 0, 4, 3, 17, 199

Offset: 0

Seiichi Manyama, Mar 03 2019

			A(6,1) = 6*Sum_{j=1..6} binomial(j,6-j)/j = 6*(1/3+3/2+1+1/6) = 18.
A(6,2) = 6*Sum_{j=1..3} binomial(j,6-2*j)/j = 6*(1/2+1/3) = 5.
Square array begins:
    2,  3, 4, 5, 6, 7, 8, 9, 10, 11, ...
    1,  0, 0, 0, 0, 0, 0, 0,  0,  0, ...
    3,  2, 0, 0, 0, 0, 0, 0,  0,  0, ...
    4,  3, 3, 0, 0, 0, 0, 0,  0,  0, ...
    7,  2, 4, 4, 0, 0, 0, 0,  0,  0, ...
   11,  5, 0, 5, 5, 0, 0, 0,  0,  0, ...
   18,  5, 3, 0, 6, 6, 0, 0,  0,  0, ...
   29,  7, 7, 0, 0, 7, 7, 0,  0,  0, ...
   47, 10, 4, 4, 0, 0, 8, 8,  0,  0, ...
   76, 12, 3, 9, 0, 0, 0, 9,  9,  0, ...

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

Columns 1-9 give A000032, A001608, A050443, A087935, A087936, A306755, A306756, A306757, A306758.

Cf. A306713, A306735.

T[0, k_] := k + 1; T[n_, k_] := n *Sum[Binomial[j, n - k*j]/j, {j, 1, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

A(0,k) = k+1 and A(n,k) = n*Sum_{j=1..floor(n/k)} binomial(j,n-k*j)/j for n > 0.

A(n,k) = (k+1)*A306713(n,k) - A306713(n-k,k) for n >= k.

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.

A017857 Expansion of 1/(1 - x^7 - x^8).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 1, 5, 10, 10, 5, 1, 0, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 2, 8, 28, 56, 70, 56, 28, 9, 10, 36, 84, 126, 126, 84, 37

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts 7 and 8. - Joerg Arndt, Jun 28 2013

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

Column k=7 of A306713.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^7-x^8))); // Vincenzo Librandi, Jun 28 2013

I:=[1,0,0,0,0,0,0,1]; [n le 8 select I[n] else Self(n-7)+Self(n-8): n in [1..70]]; // Vincenzo Librandi, Jun 28 2013

CoefficientList[Series[1 / (1 - Total[x^Range[7, 8]]), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 28 2013 *)
LinearRecurrence[{0,0,0,0,0,0,1,1},{1,0,0,0,0,0,0,1},80] (* Harvey P. Dale, Mar 19 2019 *)

x='x+O('x^66); Vec(1/(1-x^7-x^8)) \\ Altug Alkan, Oct 07 2018

a(n) = a(n-7) + a(n-8) for n > 7. - Vincenzo Librandi, Jun 28 2013

a(n) = Sum_{k=0..floor(n/7)} binomial(k,n-7*k). - Seiichi Manyama, Oct 01 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

cp's OEIS Frontend

A017847 Expansion of 1/(1 - x^6 - x^7).

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A017877 Expansion of 1/(1 - x^9 - x^10).

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A017887 Expansion of 1/(1 - x^10 - x^11).

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

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A306646 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. (k+1-x^k)/(1-x^k-x^(k+1)).

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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)).

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A017857 Expansion of 1/(1 - x^7 - x^8).

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