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.

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A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Comments

Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018

Links

Crossrefs

Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^(n k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 25}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018

A058400 Triangle of partial row sums of partition triangle A058398.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 7, 6, 5, 3, 1, 11, 10, 9, 7, 4, 1, 15, 14, 13, 11, 8, 4, 1, 22, 21, 20, 18, 15, 10, 5, 1, 30, 29, 28, 26, 23, 18, 12, 5, 1, 42, 41, 40, 38, 35, 30, 23, 14, 6, 1, 56, 55, 54, 52, 49, 44, 37, 27, 16, 6, 1, 77, 76, 75, 73, 70, 65, 58, 47, 34, 19, 7, 1, 101
Offset: 1

Views

Author

Wolfdieter Lang, Dec 11 2000

Keywords

Comments

Mirror of A026820. - Omar E. Pol, Apr 21 2012

Examples

			Triangle begins:
1;
2,   1;
3,   2,  1;
5,   4,  3,  1;
7,   6,  5,  3, 1;
11, 10,  9,  7, 4, 1;
15, 14, 13, 11, 8, 4, 1;

Crossrefs

Columns 1-3: A000041(n), A000065(n), A007042(n+1).
Cf. A008284.

Formula

a(n, m) = sum(A058398(n, k), k=m..n).

A097744 Number of ways n can be written as difference of two distinct partition numbers.

Original entry on oeis.org

2, 3, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 0, 1, 0, 2, 2, 2, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 0, 2, 2, 0, 1, 0, 1, 1, 3, 0, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 3, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 2, 1, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 23 2004

Keywords

Comments

a(A000065(n)) > 0; a(A002865(n)) > 0 for n>2;
a(A097746(n)) = 0, a(A097745(n)) > 0.

Examples

			a(27)=2: 27 = 30-3 = A000041(9)-A000041(3) =
= 42-15 = A000041(10)-A000041(7).

Crossrefs

Cf. A000041.

A317535 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} 1/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 23, 48, 106, 227, 494, 1065, 2310, 4991, 10808, 23376, 50593, 109455, 236858, 512479, 1108924, 2399418, 5191853, 11233929, 24307777, 52596430, 113806948, 246252376, 532834797, 1152933975, 2494689316, 5397944266, 11679933875, 25272740480, 54684508281, 118324934647
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 30 2018

Keywords

Comments

Invert transform of A000065.

Links

Crossrefs

Cf. A000041, A000065, A001970, A055887, A063834, A261049, A271619, A304966, A317536.

Programs

  • Maple
    seq(coeff(series(1/(1+1/(1-x)-mul(1/(1-x^k),k=1..n)), x,n+1),x,n),n=0..40); # Muniru A Asiru, Jul 30 2018
  • Mathematica
    nmax = 35; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/(1 - Sum[(PartitionsP[k] - 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, 35}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A000065(k)*x^k).

A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 11, 18, 26, 35, 49, 70, 89, 123, 164, 212, 278, 366, 460, 597, 762, 957, 1210, 1530, 1891, 2369, 2943, 3621, 4468, 5507, 6703, 8210, 10004, 12115, 14688, 17782, 21365, 25743, 30913, 36965, 44210, 52801, 62753, 74667, 88626, 104874, 124070
Offset: 0

Views

Author

Gus Wiseman, Jul 06 2023

Keywords

Comments

We use the "rounding half to even" rule, see link.

Examples

			The a(0) = 0 through a(8) = 18 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)     (7)     (8)
             (21)  (22)  (32)   (33)    (43)    (44)
                   (31)  (41)   (42)    (52)    (53)
                         (221)  (51)    (61)    (62)
                         (311)  (222)   (322)   (71)
                                (321)   (331)   (332)
                                (411)   (421)   (422)
                                (2211)  (511)   (431)
                                (3111)  (2221)  (521)
                                        (3211)  (611)
                                        (4111)  (2222)
                                                (3221)
                                                (3311)
                                                (4211)
                                                (5111)
                                                (22211)
                                                (32111)
                                                (41111)

Links

Crossrefs

Rounding-up gives A000065.
Rounding-down gives A110618, ranks A344291.
For median instead of mean we appear to have A238495.
The complement is counted by A363947, ranks A363948.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A067538 counts partitions with integer mean, ranks A316413.
A124943 counts partitions by low median, high A124944.
Cf. A002865, A098859, A241131, A327482, A363723, A363724, A363731, A363946.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]>1&]],{n,0,30}]

Formula

a(n) = A000041(n) - A363947(n).

A058717 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 1, 1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 10, 25, 18, 5, 1, 1, 14, 70, 85, 31, 6, 1, 1, 21, 217, 832, 288, 51, 7, 1
Offset: 1

Views

Author

N. J. A. Sloane, Dec 31 2000

Keywords

Examples

			1;
1,  1;
1,  2,   1;
1,  4,   3,   1;
1,  6,   9,   4,   1;
1, 10,  25,  18,   5,  1;
1, 14,  70,  85,  31,  6, 1;
1, 21, 217, 832, 288, 51, 7, 1;
...

Links

Crossrefs

Cf. A058716 (same except for border), A058710, A058711.
Row sums give A058718. Diagonals give A000065, A058719.

Extensions

Corrected and extended by Jean-François Alcover, Oct 21 2013
Reverted to original data by Jean-François Alcover, Aug 17 2022

A097128 Number of noncongruent n-dimensional integer-sided simplices with diameter 3.

Original entry on oeis.org

1, 4, 16, 56, 197, 656, 2127, 6548, 19130, 53394, 144156, 379350, 978775
Offset: 1

Views

Author

Sascha Kurz, Jul 26 2004

Keywords

Crossrefs

Cf. A000065, A097129.

Extensions

More terms from Sascha Kurz, Jul 22 2006

A097129 Number of noncongruent n-dimensional integer-sided simplices with diameter 4.

Original entry on oeis.org

1, 6, 45, 336, 3133, 31771, 329859, 3336597, 32815796
Offset: 1

Views

Author

Sascha Kurz, Jul 26 2004

Keywords

Crossrefs

Cf. A000065, A097128.

A128562 Triangle, read by rows, where T(n,k) is the coefficient of q^((n+1)*k) in the q-binomial coefficient [2*n+1, n] for n >= k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 12, 6, 1, 1, 10, 29, 29, 10, 1, 1, 14, 61, 94, 61, 14, 1, 1, 21, 120, 263, 263, 120, 21, 1, 1, 29, 222, 645, 910, 645, 222, 29, 1, 1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1, 1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2007

Keywords

Comments

Row sums equal a shifted version of A003239 (number of rooted planar trees with n non-root nodes). Column 1 is a shifted version of A000065 (-1 + number of partitions of n). Column 2 is a shifted version of A128563. This array is a variant of triangles A128545 and A047812 (Parker's partition triangle).

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   4,    1;
  1,  6,  12,    6,    1;
  1, 10,  29,   29,   10,    1;
  1, 14,  61,   94,   61,   14,    1;
  1, 21, 120,  263,  263,  120,   21,    1;
  1, 29, 222,  645,  910,  645,  222,   29,   1;
  1, 41, 392, 1468, 2724, 2724, 1468,  392,  41,  1;
  1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1;
  ...

Crossrefs

Cf. A000065 (column 1), A003239 (row sums), A128563 (column 2).
Variants are A047812 and A128545.

Programs

  • PARI
    T(n,k)=if(n

Formula

T(n,k) = [q^((n+1)*k)] Product_{j=n+1..2*n+1}(1-q^j) / Product_{j=1..n+1}(1-q^j).

Extensions

Minor edits by Petros Hadjicostas, Jun 01 2020

A141668 a(n) = tau(n) * (NumberOfPartitions(n) - 1).

Original entry on oeis.org

0, 0, 2, 4, 12, 12, 40, 28, 84, 87, 164, 110, 456, 200, 536, 700, 1150, 592, 2304, 978, 3756, 3164, 4004, 2508, 12592, 5871, 9740, 12036, 22302, 9128, 44824
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Sep 06 2008

Keywords

Links

Crossrefs

Cf. A000005, A000041.

Programs

  • Mathematica
    f[n_] = DivisorSigma[0, n]*PartitionsP[n] - DivisorSigma[0, n]; Table[f[n], {n, 1, 30}]
  • PARI
    a(n) = if (n, numdiv(n)*(numbpart(n)-1), 0); \\ Michel Marcus, Jun 11 2018

Formula

a(n) = A000005(n)*(A000041(n) - 1) = A000005(n)*A000065(n).
Previous Showing 31-40 of 44 results. Next

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

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