cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

Original entry on oeis.org

1, 0, 1, -1, 2, -4, 9, -21, 49, -112, 249, -539, 1143, -2396, 5013, -10550, 22420, -48086, 103703, -223806, 481388, -1029507, 2187944, -4625058, 9742223, -20490753, 43111808, -90840465, 191773014, -405523635, 858378825, -1817304609, 3845492204, -8129023694, 17162802918, -36191083386
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2017

Keywords

Comments

a(n) is n-th term of the Euler transform of -n + 1, 1, 1, 1, ...
Inverse zero-based binomial transform of A000041. The version for strict partitions is A380412, or A293467 up to sign. - Gus Wiseman, Feb 06 2025

Links

Crossrefs

Cf. A128566, A292463, A292541, A292613.
Cf. A000041, A002865, A053445, A275638, A275639, A275640, A275641, A275642, A275643, A275644.
Cf. A218481, A294466, A095051.
Cf. A266232, A294467, A293467, A294468.
Row n=0 of A175804 (row n=1 is A320590 except first term).
Cf. A000009, A008284, A087897, A116608, A129519, A225486, A320591, A377056, A378622.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0,
      combinat[numbpart](n), b(n, k-1)-b(n-1, k-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2024
  • Mathematica
    Table[SeriesCoefficient[(1 - q)^n / Product[(1 - q^j), {j, 1, n}], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[(1 - q)^n QPochhammer[q^(1 + n), q]/QPochhammer[q, q], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[1/QFactorial[n, q], {q, 0, n}], {n, 0, 35}]
Table[Differences[PartitionsP[Range[0, n]], n], {n, 0, 35}] // Flatten
Table[Sum[(-1)^j*Binomial[n, j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 06 2017 *)

Formula

a(n) = [q^n] 1/((1 + q)*(1 + q + q^2)*...*(1 + q + ... + q^(n-1))).
a(n) = Sum_{j=0..n} (-1)^j * binomial(n, j) * A000041(n-j). - Vaclav Kotesovec, Oct 06 2017
a(n) ~ (-1)^n * 2^(n - 3/2) * exp(Pi*sqrt(n/12) + Pi^2/96) / (sqrt(3)*n). - Vaclav Kotesovec, May 07 2018

A275639 Expansion of (1-q)^k/Product_{j=1..k} (1-q^j) for k=5.

Original entry on oeis.org

1, -4, 7, -7, 5, -4, 4, -4, 5, -7, 8, -8, 9, -11, 12, -11, 9, -8, 9, -11, 13, -15, 16, -15, 14, -15, 16, -15, 14, -15, 17, -19, 21, -22, 21, -19, 18, -19, 21, -22, 22, -23, 25, -26, 26, -26, 25, -23, 23, -26, 29, -30, 30, -30, 30, -30, 30, -30, 30, -30, 31, -34, 37, -37, 35, -34, 34, -34, 35
Offset: 0

Views

Author

N. J. A. Sloane, Aug 09 2016

Keywords

Links

Crossrefs

Cf. A275638.

Programs

  • PARI
    Vec(1/((1+x)^2*(1+x^2)*(1+x+x^2)*(1+x+x^2+x^3+x^4)) + O(x^100)) \\ Colin Barker, Aug 11 2016

Formula

Equivalent g.f.: 1 / ((1+x)^2*(1+x^2)*(1+x+x^2)*(1+x+x^2+x^3+x^4)). - Colin Barker, Aug 10 2016
a(n) = -4*a(n-1) - 9*a(n-2) - 15*a(n-3) - 20*a(n-4) - 22*a(n-5) - 20*a(n-6) - 15*a(n-7) - 9*a(n-8) - 4*a(n-9) - a(n-10). - Ilya Gutkovskiy, Aug 10 2016

A275640 Expansion of (1-q)^k/Product_{j=1..k} (1-q^j) for k=6.

Original entry on oeis.org

1, -5, 11, -14, 12, -9, 9, -13, 20, -26, 27, -25, 26, -33, 43, -49, 47, -42, 43, -53, 67, -77, 78, -73, 72, -82, 98, -108, 107, -102, 104, -118, 138, -151, 150, -142, 141, -155, 178, -194, 194, -187, 189, -206, 230, -246, 245, -235, 235, -255, 285, -305, 305, -295, 295, -315, 345, -365, 365
Offset: 0

Views

Author

N. J. A. Sloane, Aug 09 2016

Keywords

Links

Crossrefs

Cf. A275638.

Programs

  • Mathematica
    CoefficientList[Series[1/((1+x)^3(1-x+x^2)(1+x^2)(1+x+x^2)^2(1+ x+x^2+ x^3+x^4)),{x,0,100}],x] (* or *) LinearRecurrence[{-5,-14,-29,-49,-71,-90,-101,-101,-90,-71,-49,-29,-14,-5,-1},{1,-5,11,-14,12,-9,9,-13,20,-26,27,-25,26,-33,43},100] (* Harvey P. Dale, Mar 14 2023 *)

Formula

Equivalent g.f.: 1 / ((1+x)^3*(1-x+x^2)*(1+x^2)*(1+x+x^2)^2*(1+x+x^2+x^3+x^4)). - Colin Barker, Aug 10 2016

A275641 Expansion of (1-q)^k/Product_{j=1..k} (1-q^j) for k=7.

Original entry on oeis.org

1, -6, 16, -25, 26, -21, 18, -21, 27, -30, 28, -26, 30, -41, 55, -65, 66, -61, 59, -66, 79, -89, 90, -85, 84, -95, 114, -127, 126, -119, 121, -138, 161, -175, 174, -166, 164, -175, 195, -211, 213, -207, 210, -231, 261, -281, 280, -267, 263, -280, 309, -329, 329, -320, 323, -347, 380, -401, 401
Offset: 0

Views

Author

N. J. A. Sloane, Aug 09 2016

Keywords

Links

Crossrefs

Cf. A275638.

Programs

  • Mathematica
     CoefficientList[Series[1/((1 + x)^3 (1 - x + x^2) (1 + x^2)(1 + x + x^2)^2 (1 + x + x^2 + x^3 + x^4) (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 80}], x] (* Vincenzo Librandi, Feb 04 2017 *)
  • PARI
    Vec(1/((1+x)^3*(1-x+x^2)*(1+x^2)*(1+x+x^2)^2*(1+x+x^2+x^3+x^4)*(1+x+x^2+x^3+x^4+x^5+x^6)) + O(x^100)) \\ Colin Barker, Aug 11 2016

Formula

An equivalent but more complicated g.f.: 1 / ((1+x)^3*(1-x+x^2)*(1+x^2)*(1+x+x^2)^2*(1+x+x^2+x^3+x^4)*(1+x+x^2+x^3+x^4+x^5+x^6)). - Colin Barker, Aug 10 2016

A275642 Expansion of (1-q)^k/Product_{j=1..k} (1-q^j) for k=8.

Original entry on oeis.org

1, -7, 22, -41, 51, -47, 39, -39, 49, -64, 80, -95, 107, -118, 135, -159, 180, -191, 200, -220, 252, -286, 314, -334, 349, -370, 409, -461, 505, -531, 554, -593, 648, -706, 758, -801, 835, -870, 924, -999, 1072, -1126, 1175, -1242, 1327, -1412, 1485, -1546, 1602, -1669, 1764, -1880, 1985, -2061
Offset: 0

Views

Author

N. J. A. Sloane, Aug 09 2016

Keywords

Links

Crossrefs

Cf. A275638.

A275643 Expansion of (1-q)^k/Product_{j=1..k} (1-q^j) for k=12.

Original entry on oeis.org

1, -11, 56, -175, 376, -592, 719, -722, 678, -696, 815, -988, 1143, -1254, 1374, -1602, 1986, -2440, 2794, -2971, 3095, -3378, 3906, -4562, 5147, -5568, 5937, -6487, 7336, -8317, 9111, -9601, 10048, -10852, 12146, -13656, 14954, -15853, 16601, -17692, 19402, -21465, 23298, -24631, 25843, -27577
Offset: 0

Views

Author

N. J. A. Sloane, Aug 09 2016

Keywords

Links

Crossrefs

Cf. A275638.

A275644 Expansion of (1-q)^k/Product_{j=1..k} (1-q^j) for k=16.

Original entry on oeis.org

1, -15, 106, -469, 1457, -3381, 6099, -8829, 10624, -11208, 11274, -11858, 13447, -15709, 18001, -20090, 22420, -25667, 29965, -34627, 38835, -42687, 47352, -53944, 62270, -70875, 78399, -84913, 91988, -101370, 113336, -126336, 138584, -150174, 163347, -180341, 200990, -222634, 242515
Offset: 0

Views

Author

N. J. A. Sloane, Aug 09 2016

Keywords

Links

Crossrefs

Cf. A275638.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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