A017887 -id:A017887 - cp's OEIS frontend

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).

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

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).

A017890 Expansion of 1/(1-x^10-x^11-x^12-x^13-x^14).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 7, 7, 11, 20, 35, 52, 68, 80, 85, 80, 69, 57, 50, 55, 80, 125, 186, 255, 320, 365, 382, 371, 341, 311, 311, 367, 496, 701, 966, 1251, 1508, 1693, 1779, 1770, 1716, 1701, 1826

Offset: 0

N. J. A. Sloane

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

Cf. A017887.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!(1/(1-x^10-x^11-x^12-x^13-x^14))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[10, 14]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017890_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^15) ).list()
A017890_list(80) # G. C. Greubel, Nov 06 2024

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

A017891 Expansion of 1/(1-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, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 2, 3, 6, 10, 15, 21, 25, 27, 27, 25, 22, 19, 20, 26, 38, 57, 80, 104, 125, 140, 147, 145, 140, 139, 150, 182, 240, 325, 430, 544, 653, 741, 801, 836, 861, 903, 996, 1176, 1466, 1871, 2374, 2933, 3494, 4005, 4436

Offset: 0

N. J. A. Sloane

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

Cf. A017887.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!(1/(1-x^10-x^11-x^12-x^13-x^14-x^15))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1/(1 - Total[x^Range[10, 15]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017891_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^16) ).list()
A017891_list(80) # G. C. Greubel, Nov 06 2024

a(n) = 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

A017892 Expansion of 1/(1 - 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, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 4, 5, 7, 10, 15, 21, 28, 33, 36, 37, 37, 37, 38, 41, 50, 66, 90, 119, 150, 180, 207, 229, 246, 259, 276, 306, 359, 441, 554, 696, 862, 1041, 1221, 1390, 1547, 1703, 1882, 2116, 2441, 2891, 3494, 4259, 5174, 6205

Offset: 0

N. J. A. Sloane

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

Cf. A017887.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!(1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16))); // Vincenzo Librandi, Jul 01 2013

a[0]:= 1:
for i from 1 to 9 do a[i]:= 0 od:
for i from 10 to 15 do a[i]:= 1 od:
for i from 16 to 1000 do a[i]:= add(a[j],j=i-16 .. i-10) od:
seq(a[i],i=0..1000); # Robert Israel, Aug 15 2014

(* From Harvey P. Dale, Mar 04 2013: (Start) *)
CoefficientList[Series[1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16),{x,0,80}],x]
LinearRecurrence[{0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}, {1,0,0,0,0,0,0,0,0,0,1, 1,1,1,1,1},70] (* End *)
CoefficientList[Series[1/(1 - Total[x^Range[10, 16]]), {x,0,80}], x] (* Vincenzo Librandi, Jul 01 2013 *)

my(x='x+O('x^80)); Vec(1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16)) \\ Altug Alkan, Oct 04 2018

def A017892_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^17) ).list()
A017892_list(80) # G. C. Greubel, Nov 06 2024

a(n) = a(n-10) + a(n-11) + a(n-12) + a(n-13) + a(n-14) + a(n-15) + a(n-16); 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)=0, a(10)=1, a(11)=1, a(12)=1, a(13)=1, a(14)=1, a(15)=1. - Harvey P. Dale, Mar 04 2013

A017893 Expansion of 1/(1-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, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 6, 7, 9, 12, 16, 21, 28, 36, 42, 46, 49, 52, 56, 62, 71, 84, 105, 135, 171, 210, 250, 290, 330, 371, 414, 462, 525, 614, 736, 894, 1088, 1316, 1575, 1862, 2171, 2498, 2852, 3256, 3742, 4346, 5104, 6049, 7210, 8610

Offset: 0

N. J. A. Sloane

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

Cf. A017887.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!(1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17))); // Vincenzo Librandi, Jul 01 2013

a:= n-> (Matrix(17, (i, j)-> if (i=j-1) or (j=1 and i in [$10..17]) then 1 else 0 fi)^n)[1, 1]: seq(a(n), n=0..80);  # Alois P. Heinz, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[10, 17]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1},{1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1},80] (* Harvey P. Dale, Dec 02 2024 *)

def A017893_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^18) ).list()
A017893_list(80) # G. C. Greubel, Nov 06 2024

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

A017894 Expansion of 1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 8, 9, 11, 14, 18, 23, 29, 36, 45, 52, 58, 64, 71, 80, 92, 108, 129, 156, 193, 237, 286, 339, 396, 458, 527, 606, 699, 810, 951, 1130, 1352, 1620, 1936, 2302, 2721, 3198, 3741, 4358, 5072, 5916, 6929, 8153, 9631

Offset: 0

N. J. A. Sloane

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

Cf. A017887.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!(1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18))); // Vincenzo Librandi, Jul 01 2013

a:= n-> (Matrix(18, (i, j)-> if (i=j-1) or (j=1 and i in [$10..18]) then 1 else 0 fi)^n)[1, 1]: seq(a(n), n=0..80);  # Alois P. Heinz, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[10, 18]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)

def A017894_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^19) ).list()
A017894_list(80) # G. C. Greubel, Nov 06 2024

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

A017895 Expansion of 1/(1-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, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 13, 16, 20, 25, 31, 38, 46, 55, 64, 73, 83, 95, 110, 129, 153, 183, 220, 265, 319, 381, 451, 530, 620, 724, 846, 991, 1165, 1375, 1630, 1938, 2306, 2741, 3251, 3846, 4539, 5347, 6292, 7402, 8713, 10270

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19. - 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 (0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1).

Cf. A017887.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!(1/(1-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[10, 19]]), {x, 0, 70}], x] (* Vincenzo Librandi Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1},{1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1},80] (* Harvey P. Dale, Apr 07 2025 *)

def A017895_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^20) ).list()
A017895_list(81) # G. C. Greubel, Nov 08 2024

a(n) = 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

A017896 Expansion of 1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18-x^19-x^20).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 18, 22, 27, 33, 40, 48, 57, 68, 79, 92, 107, 125, 147, 174, 207, 247, 295, 353, 420, 499, 591, 698, 823, 970, 1144, 1351, 1598, 1894, 2246, 2666, 3165, 3756, 4454, 5277, 6247, 7391, 8742, 10341, 12234

Offset: 0

N. J. A. Sloane

Number of compositions (ordered partitions) of n into parts 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20. - 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 (0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1).

Cf. A017887.

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!(1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18-x^19-x^20))); // Vincenzo Librandi, Jul 01 2013

a:= n-> (Matrix(20, (i, j)-> if (i=j-1) or (j=1 and i in [$10..20]) then 1 else 0 fi)^n)[1, 1]: seq(a(n), n=0..80);  # Alois P. Heinz, Aug 04 2008

CoefficientList[Series[1/(1 -Total[x^Range[10, 20]]), {x,0,80}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1},{1,0,0,0,0,0,0,0, 0,0,1,1,1,1,1,1,1,1,1,1}, 81] (* Harvey P. Dale, Oct 21 2016 *)

def A017896_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^21) ).list()
A017896_list(81) # G. C. Greubel, Nov 08 2024

a(n) = 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) +a(n-20), for n>19. - Vincenzo Librandi, Jul 01 2013

A017888 Expansion of 1/(1 - x^10 - x^11 - x^12).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

Cf. A017887, A017889, A017890, A017891, A017892, A017893, A017894, A017895, A017896.

m:=80; R:=PowerSeriesRing(Integers(), m);
Coefficients(R!(1/(1-x^10-x^11-x^12))); // Vincenzo Librandi, Jul 01 2013

CoefficientList[Series[1 / (1 - Total[x^Range[10, 12]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 01 2013 *)

my(x='x+O('x^80)); Vec(1/(1-x^10-x^11-x^12)) \\ Altug Alkan, Oct 04 2018

def A017888_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-x-x^10+x^(13)) ).list()
A017888_list(80) # G. C. Greubel, Sep 25 2024

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

cp's OEIS Frontend

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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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)).

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A017890 Expansion of 1/(1-x^10-x^11-x^12-x^13-x^14).

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A017891 Expansion of 1/(1-x^10-x^11-x^12-x^13-x^14-x^15).

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A017892 Expansion of 1/(1 - x^10 - x^11 - x^12 - x^13 - x^14 - x^15 - x^16).

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A017893 Expansion of 1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17).

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A017894 Expansion of 1/(1-x^10-x^11-x^12-x^13-x^14-x^15-x^16-x^17-x^18).

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