A017877 -id:A017877 - cp's OEIS frontend

A017885 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18).

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 17, 21, 26, 32, 39, 47, 57, 67, 79, 93, 110, 131, 157, 189, 228, 276, 332, 399, 478, 571, 681, 812, 969, 1158, 1387, 1662, 1994

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. - 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 (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1).

Cf. A017877, A017878, A017879, A017880, A017881, A017882, A017883, A017884, A017886.

m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 18]]), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017885_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(19)) ).list()
A017885_list(70) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) +a(n-16) +a(n-17) +a(n-18), for n>17. - Vincenzo Librandi, Jul 01 2013

A017886 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18-x^19).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 19, 23, 28, 34, 41, 49, 59, 72, 86, 102, 122, 146, 175, 210, 252, 303, 366, 441, 529, 635, 762, 914, 1096, 1314, 1576, 1893, 2275

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts 9, 10, 11, ..., 19. - Joerg Arndt, Oct 12 2014

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

Cf. A017877, A017878, A017879, A017880, A017881, A017882, A017883, A017884, A017885, A017887.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18-x^19))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 19]]), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017886_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(20)) ).list()
A017886_list(70) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) +a(n-16) +a(n-17) +a(n-18) +a(n-19) for n>18. - Vincenzo Librandi, Jul 01 2013

a(n) = a(n-1) +a(n-9) -a(n-20) for n>19. - Tani Akinari, Sep 29 2014

A017878 Expansion of 1/(1-x^9-x^10-x^11).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 0, 1, 3, 6, 7, 6, 3, 1, 0, 0, 1, 4, 10, 16, 19, 16, 10, 4, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 6, 7, 21, 50, 90, 126, 141, 126, 90, 51, 28, 34, 78, 161

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 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 (0,0,0,0,0,0,0,0,1,1,1).

Cf. A017877, A017879, A017880, A017881, A017882, A017883, A017884, A017885, A017886.

m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 11]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1},{1,0,0,0,0,0,0,0,0,1,1},70] (* Harvey P. Dale, May 25 2023 *)

def A017878_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(12)) ).list()
A017878_list(80) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) +a(n-10) +a(n-11) for n>10. - Vincenzo Librandi, Jul 01 2013

A017879 Expansion of 1/(1-x^9-x^10-x^11-x^12).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 3, 6, 10, 12, 12, 10, 6, 3, 2, 4, 10, 20, 31, 40, 44, 40, 31, 21, 15, 19, 36, 65, 101, 135, 155, 155, 136, 107, 86, 91, 135, 221, 337, 456, 546

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 10, 11 and 12. - 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 (0,0,0,0,0,0,0,0,1,1,1,1).

Cf. A017877, A017878, A017880, A017881, A017882, A017883, A017884, A017885, A017886.

m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1/(1-x^9 -x^10 -x^11 -x^12), {x,0,70}], x] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1,1}, {1,0,0,0,0,0,0,0,0,1,1,1}, 70] (* Harvey P. Dale, Apr 29 2013 *)
CoefficientList[Series[1/(1 - Total[x^Range[9, 12]]), {x,0,70}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017879_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(13)) ).list()
A017879_list(85) # G. C. Greubel, Sep 25 2024

a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=1, a(10)=1, a(11)=1; for n>11, a(n) = a(n-9)+a(n-10)+a(n-11)+a(n-12). - Harvey P. Dale, Apr 29 2013

A017880 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 18, 19, 18, 15, 11, 10, 13, 21, 35, 52, 68, 80, 85, 81, 73, 67, 70, 90, 131, 189, 256, 320, 366, 387, 386, 376, 381, 431, 547

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12 and 13. - 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 (0,0,0,0,0,0,0,0,1,1,1,1,1).

Cf. A017877, A017878, A017879, A017881, A017882, A017883, A017884, A017885, A017886.

m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 13]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1,1,1},{1,0,0,0,0,0,0,0,0,1,1,1,1},70] (* Harvey P. Dale, Apr 03 2018 *)

def A017880_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(14)) ).list()
A017880_list(80) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) for n>12. - Vincenzo Librandi, Jul 01 2013

A017881 Expansion of 1/(1 - x^9 - x^10 - x^11 - x^12 - x^13 - x^14).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 3, 4, 6, 10, 15, 21, 25, 27, 27, 26, 25, 25, 30, 41, 59, 81, 104, 125, 141, 151, 155, 160, 174, 206, 261, 340, 440, 551, 661, 757, 836, 906, 987

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13 and 14. - 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 (0,0,0,0,0,0,0,0,1,1,1,1,1,1).

Cf. A017877, A017878, A017879, A017880, A017882, A017883, A017884, A017885, A017886.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (1-x)/(1-x-x^9+x^(15)) )); // G. C. Greubel, Sep 25 2024

CoefficientList[Series[1/(1-Total[x^Range[9,14]]),{x,0,60}],x] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,1,1,1,1,1,1},{1,0,0,0,0,0,0,0,0,1,1,1,1,1}, 60] (* Harvey P. Dale, Feb 27 2012 *)

def A017881_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(15)) ).list()
A017881_list(80) # G. C. Greubel, Sep 25 2024

a(0)=1, a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=a(7)=a(8)=0, a(9)=a(10)=a(11)=a(12)= a(13)=1, a(n) = a(n-9) + a(n-10) + a(n-11) + a(n-12) + a(n-13) + a(n-14). - Harvey P. Dale, Feb 27 2012

A017882 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 5, 6, 8, 11, 15, 21, 28, 33, 36, 38, 40, 43, 48, 56, 71, 94, 122, 152, 182, 211, 239, 266, 294, 332, 390, 474, 586, 725, 888, 1071, 1266, 1466

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13, 14 and 15. - 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 (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1).

Cf. A017877, A017878, A017879, A017880, A017881, A017883, A017884, A017885, A017886.

m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15)));  // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 15]]),{x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017882_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(16)) ).list()
A017882_list(80) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) for n>14. - Vincenzo Librandi, Jul 01 2013

A017883 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 7, 8, 10, 13, 17, 22, 28, 36, 42, 47, 52, 58, 66, 77, 92, 112, 141, 176, 215, 257, 302, 351, 406, 470, 546, 645, 774, 937, 1136, 1372, 1646

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13, 14, 15 and 16. - 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 (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1).

Cf. A017877, A017878, A017879, A017880, A017881, A017882, A017884, A017885, A017886.

m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 16]]), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017883_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(17)) ).list()
A017883_list(65) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) +a(n-16) for n>15. - Vincenzo Librandi, Jul 01 2013

A017884 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 12, 15, 19, 24, 30, 37, 45, 53, 61, 70, 81, 95, 113, 136, 165, 201, 245, 296, 354, 420, 496, 585, 691, 819, 975, 1167, 1402, 1686, 2025

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 9, 10, 11, 12, 13, 14, 15, 16 and 17. - 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 (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1).

Cf. A017877, A017878, A017879, A017880, A017881, A017882, A017883, A017885, A017886.

m:=70; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[9, 17]]), {x, 0, 60}], x] (* Harvey P. Dale, Sep 12 2012 *)

def A017884_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^9+x^(18)) ).list()
A017884_list(70) # G. C. Greubel, Sep 25 2024

a(n) = a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15) +a(n-16) +a(n-17) for n>16. - Vincenzo Librandi, Jul 01 2013

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).

cp's OEIS Frontend

A017885 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18).

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A017886 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18-x^19).

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A017878 Expansion of 1/(1-x^9-x^10-x^11).

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A017879 Expansion of 1/(1-x^9-x^10-x^11-x^12).

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A017880 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13).

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A017881 Expansion of 1/(1 - x^9 - x^10 - x^11 - x^12 - x^13 - x^14).

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A017882 Expansion of 1/(1-x^9-x^10-x^11-x^12-x^13-x^14-x^15).

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