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.

Previous Showing 11-20 of 23 results. Next

A299166 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).

Original entry on oeis.org

1, 1, 2, 6, 17, 48, 132, 365, 1003, 2759, 7583, 20843, 57283, 157442, 432719, 1189317, 3268818, 8984318, 24693343, 67869557, 186539251, 512702559, 1409161449, 3873076007, 10645137706, 29258128633, 80415877302, 221022792843, 607480469466, 1669658209311, 4589050472041
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A255961.
Cf. A000219, A067687, A299105, A299106, A299108, A299162, A299164, A299167.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
       b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(n-j, j), j=0..n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 04 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - x Product[1/(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A000219(k-1)*a(n-k).

A299167 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)^k).

Original entry on oeis.org

1, 1, 2, 5, 14, 36, 94, 243, 628, 1619, 4178, 10776, 27793, 71682, 184879, 476832, 1229830, 3171942, 8180989, 21100215, 54421187, 140361900, 362018270, 933709453, 2408202606, 6211182512, 16019743522, 41317765457, 106565859669, 274852289679, 708892898170, 1828360759013, 4715667307920
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A277938.
Cf. A026007, A067687, A299105, A299106, A299108, A299162, A299164, A299166.

Programs

  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 - x Product[(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A026007(k-1)*a(n-k).

A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

Original entry on oeis.org

1, 1, 0, -3, -6, -4, 12, 39, 52, -9, -186, -392, -285, 610, 2291, 3200, -150, -10626, -23487, -18841, 32957, 134848, 198246, 13961, -605248, -1409604, -1234474, 1744213, 7898753, 12209679, 2161666, -34344627, -84393284, -79993042, 90692470, 461463974, 749309529, 207447895, -1939084232
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A276554.
Cf. A067687, A073592, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299212.

Programs

  • Maple
    N:= 100: # for a(0)..a(N)
S:= series(1/(1-x*mul((1-x^k)^k,k=1..N)),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Feb 05 2023
  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A073592(k-1)*a(n-k).

A060850 Array of the coefficients A(n,k) in the expansion of Product_{i>=1} 1/(1-x^i)^n = Sum_{k>=0} A(n,k)*x^k, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 7, 1, 6, 20, 40, 51, 36, 11, 1, 7, 27, 65, 105, 108, 65, 15, 1, 8, 35, 98, 190, 252, 221, 110, 22, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 1, 11, 65, 255
Offset: 1

Views

Author

Bo T. Ahlander (ahlboa(AT)isk.kth.se), May 03 2001

Keywords

Comments

Table read by antidiagonals: entry (n,k) gives number of partitions of n objects into parts of k kinds. - Franklin T. Adams-Watters, Dec 28 2006

Examples

			Table (row k, k >= 0: number of partitions of n, n >= 0, into parts of k kinds):
Array begins:
=======================================================================
k\n| 0   1   2    3     4     5      6       7       8       9       10
---|-------------------------------------------------------------------
1  | 1   1   2    3     5     7     11      15      22      30       42
2  | 1   2   5   10    20    36     65     110     185     300      481
3  | 1   3   9   22    51   108    221     429     810    1479     2640
4  | 1   4  14   40   105   252    574    1240    2580    5180    10108
5  | 1   5  20   65   190   506   1265    2990    6765   14725    31027
6  | 1   6  27   98   315   918   2492    6372   15525   36280    81816
7  | 1   7  35  140   490  1547   4522   12405   32305   80465   192899
8  | 1   8  44  192   726  2464   7704   22528   62337  164560   417140
9  | 1   9  54  255  1035  3753  12483   38709  113265  315445   841842
10 | 1  10  65  330  1430  5512  19415   63570  195910  573430  1605340
11 | 1  11  77  418  1925  7854  29183  100529  325193  997150  2919411
  ...
Triangle (row n, n >= 0: number of partitions of n into parts of n - k kinds, 0 <= k <= n) (antidiagonals of above table) (parenthesized last term on each row, which would correspond to row k = 0 in above table)
Triangle begins: (column k: n - k kinds of parts)
===================================
n\k| 0   1   2   3   4   5   6   7
---+-------------------------------
0  |(1)
1  | 1, (0)
2  | 1,  1, (0)
3  | 1,  2,  2, (0)
4  | 1,  3,  5,  3, (0)
5  | 1,  4,  9, 10,  5, (0)
6  | 1,  5, 14, 22, 20,  7, (0)
7  | 1,  6, 20, 40, 51, 36, 11, (0)
  ...

Crossrefs

Cf. A067687 (table antidiagonal sums, triangle row sums).
Rows (table), diagonals (triangle): A000041, A000712, A000716, A023003-A023021, A006922.
Columns (table, triangle): A000012, A001477, A000096, A006503, A006504.

Programs

  • Mathematica
    t[n_, k_] := CoefficientList[ Series[ Product[1/(1 - x^i)^n, {i, k}], {x, 0, k}], x][[k]]; (* Robert G. Wilson v, Aug 08 2018 *)
t[n_, k_]; = IntegerPartitions[n, {k}]; Table[ t[n - k + 1, k], {n, 12}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 08 2018 *)

Formula

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=1..n} A000041(k-1)*A(n-k;x)*x^(k-1), A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004

Extensions

More terms from Vladeta Jovovic, Jan 02 2004

A299209 Expansion of 1/(1 - x*Product_{k>=1} (1 - k*x^k)).

Original entry on oeis.org

1, 1, 0, -3, -6, -5, 11, 37, 59, 13, -155, -402, -415, 263, 1981, 3748, 2289, -6643, -22642, -31322, -187, 99040, 229410, 216823, -230029, -1223267, -2097812, -955237, 4468902, 13393758, 16752461, -3891704, -62382597, -131974181, -106680562, 173622424, 741553622, 1163057561, 329176545
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297323.
Cf. A022661, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299210, A299211, A299212.

Programs

  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - x Product[1 - k x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022661(k-1)*a(n-k).

A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

Original entry on oeis.org

1, 1, 0, -2, -5, -3, 5, 20, 27, 17, -53, -152, -192, 31, 576, 1110, 694, -1297, -4519, -6160, -1107, 13665, 31914, 30643, -19339, -119260, -196142, -103318, 289543, 859631, 1062684, 13710, -2690348, -5675946, -4940757, 4167527, 21343918, 33874107, 16524162, -51704908, -150454546
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297325.
Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022693(k-1)*a(n-k).

A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

Original entry on oeis.org

1, 1, 0, -2, -5, -4, 4, 21, 35, 23, -47, -165, -239, -78, 479, 1273, 1508, -138, -4429, -9451, -8845, 6207, 37937, 67123, 45144, -83355, -308078, -455109, -166872, 873799, 2393041, 2916869, -73472, -8133572, -17828640, -17294146, 10383571, 70275162, 127401305, 90368779, -147825714
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A279928.
Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k).

A318581 Expansion of 1/(1 + x*Product_{k>=1} 1/(1 - x^k)).

Original entry on oeis.org

1, -1, 0, -1, 0, -1, 1, -1, 3, -1, 5, -2, 7, -7, 9, -16, 11, -29, 20, -46, 45, -66, 94, -95, 175, -161, 294, -307, 458, -594, 715, -1096, 1193, -1891, 2132, -3106, 3916, -5063, 7083, -8484, 12347, -14770, 20867, -26310, 34898, -46771, 58967, -81665, 101680, -139951, 178094, -237620
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

Examples

			G.f. = 1 - x - x^3 - x^5 + x^6 - x^7 + 3*x^8 - x^9 + 5*x^10 - 2*x^11 + 7*x^12 - 7*x^13 + ...

Links

Crossrefs

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318582, A331484.
Cf. A000041, A010815.

Programs

  • Maple
    seq(coeff(series((1+x*mul((1-x^k)^(-1),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 55); # Muniru A Asiru, Aug 30 2018
  • Mathematica
    nmax = 51; CoefficientList[Series[1/(1 + x Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[PartitionsP[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 51}]

Formula

G.f.: 1/(1 + x*Sum_{k>=0} A000041(k)*x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A000041(k-1)*a(n-k).

A318582 Expansion of 1/(1 + x*Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, 0, 1, 0, 0, 0, 1, -1, 0, 1, -3, 2, -1, -3, 4, -4, 0, 3, -5, 4, 0, -2, 4, -1, 1, 0, 3, -2, 0, 6, -11, 9, -1, -13, 18, -17, 1, 13, -23, 17, -4, -8, 13, -8, 7, -6, 15, -10, -3, 33, -50, 42, 0, -56, 85, -72, 6, 59, -100, 75, -23, -34, 53, -44, 35
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

Examples

			G.f. = 1 - x - x^4 + x^5 - x^6 + x^8 - x^9 + x^10 + x^13 + x^17 - x^18 + x^20 - 3*x^21 + ...

Links

Crossrefs

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A331484.
Cf. A000009, A081362.

Programs

  • Maple
    a:=series(1/(1+x*mul(1+x^k,k=1..100)),x=0,76): seq(coeff(a,x,n),n=0..75); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 75; CoefficientList[Series[1/(1 + x Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

G.f.: 1/(1 + x*Sum_{k>=0} A000009(k)*x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A000009(k-1)*a(n-k).

A331484 Expansion of 1/(1 + x*Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, -1, 2, -2, 3, -3, 3, -2, -1, 5, -13, 22, -36, 51, -68, 82, -86, 75, -31, -52, 201, -421, 732, -1125, 1575, -2024, 2344, -2370, 1807, -327, -2532, 7210, -14128, 23486, -35027, 47799, -59594, 66717, -63246, 41012, 10696, -104335, 252653, -465825, 746343
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2020

Keywords

Links

Crossrefs

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582.
Cf. A010815.

Programs

  • Mathematica
    m = 44; CoefficientList[Series[1/(1 + x*Product[1 - x^k, {k, 1, m}]), {x, 0, m}], x] (* Amiram Eldar, May 05 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/(1+x*prod(k=1, N, 1-x^k)))

Formula

a(0) = 1, a(n) = -Sum_{k=1..n} A010815(k-1)*a(n-k) for n > 0.
Previous Showing 11-20 of 23 results. Next

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

Content is available under The OEIS End-User License Agreement.