A006503 -id:A006503 - cp's OEIS frontend

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.

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

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

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

			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)
  ...

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.

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 *)

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

More terms from Vladeta Jovovic, Jan 02 2004

A185783 Second accumulation array of A185780, by antidiagonals.

Original entry on oeis.org

1, 6, 3, 20, 20, 6, 50, 70, 44, 10, 105, 180, 160, 80, 15, 196, 385, 420, 300, 130, 21, 336, 728, 910, 800, 500, 196, 28, 540, 1260, 1736, 1750, 1350, 770, 280, 36, 825, 2040, 3024, 3360, 2975, 2100, 1120, 384, 45, 1210, 3135, 4920, 5880, 5740, 4655, 3080, 1560, 510, 55, 1716, 4620, 7590, 9600, 10080, 9016, 6860, 4320, 2100, 660, 66, 2366, 6578, 11220, 14850, 16500, 15876, 13328, 9660, 5850, 2750, 836, 78

Offset: 1

Clark Kimberling, Feb 03 2011

See A144112 and A185780.

			Northwest corner:
1....6....20....50....105
3....20...70....180...385
6....44...160...420...910
10...80...300...800...1750

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185780.
Row 1: A002415 (4-dimensional pyramidal numbers).
Columns 1 to 3: A000217, 2*A006503, 10*A005581.

(See A185780.)
f[n_, k_] := Binomial[k + 2, 3]*Binomial[n + 1, 2]*(k*n - n + 2*k + 4)/6; Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = C(k+2,3)*C(n+1,2)*(k*n-n+2*k+4)/6, k>=1, n>=1.

A027378 Expansion of (1+x^2-x^3)/(1-x)^4.

Original entry on oeis.org

1, 4, 11, 23, 41, 66, 99, 141, 193, 256, 331, 419, 521, 638, 771, 921, 1089, 1276, 1483, 1711, 1961, 2234, 2531, 2853, 3201, 3576, 3979, 4411, 4873, 5366, 5891, 6449, 7041, 7668, 8331, 9031, 9769, 10546

Offset: 0

N. J. A. Sloane

If Y is a 3-subset of an n-set X then, for n>=4, a(n-4) is the number of (n-3)-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006416, A006503, A034857, A116721.
Appears to be first differences of A252814.
First differences at A027379 (omitting first term).

[(n^3 +9*n^2 +8*n +6)/6: n in [0..50]]; // G. C. Greubel, Jul 30 2022

CoefficientList[Series[(1+x^2-x^3)/(1-x)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,11,23},50] (* Harvey P. Dale, May 17 2021 *)

[(n^3 +9*n^2 +8*n +6)/6 for n in (0..50)] # G. C. Greubel, Jul 30 2022

a(n) = binomial(n+4, 3) - 3*(n+1). - Milan Janjic, Dec 28 2007 [Correction by Mathew Englander, Feb 03 2022]
a(n) = A006503(n) + 1 = A034857(n) + 5 = A116721(n+2) - 1 = A006416(n+1) + 3. - Mathew Englander, Feb 03 2022
E.g.f.: (1/6)*(6 + 18*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 30 2022

A139798 Coefficient of x^5 in (1-x-x^2)^(-n).

Original entry on oeis.org

8, 38, 111, 256, 511, 924, 1554, 2472, 3762, 5522, 7865, 10920, 14833, 19768, 25908, 33456, 42636, 53694, 66899, 82544, 100947, 122452, 147430, 176280, 209430, 247338, 290493, 339416, 394661, 456816, 526504, 604384, 691152, 787542

Offset: 1

Sergio Falcon, May 22 2008

The coefficient of x^5 in (1-x-x^2)^(-n) is the coefficient of x^5 in (1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5)^n. Using the multinomial theorem one then finds that a(n) = n(n+1)(n+2)(n^2 + 27n + 132)/5!

The inverse binomial transform yields 8,30,43,29,9,1,0,0,... (0 continued) - R. J. Mathar, May 23 2008

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000096, A006503, A006504.

a[n_] := n(n + 1)(n + 2)(n^2 + 27n + 132)/5! Do[Print[n, " ", a[n]], {n, 1, 25}]
LinearRecurrence[{6,-15,20,-15,6,-1},{8,38,111,256,511,924},40] (* Harvey P. Dale, Oct 13 2015 *)

a(n)=binomial(n+2,3)*(n^2+27*n+132)/20 \\ Charles R Greathouse IV, Jul 29 2011

a(n) = n(n+1)(n+2)(n^2 + 27n + 132)/5!

O.g.f.: x(3x-4)(x-2)/(1-x)^6. - R. J. Mathar, May 23 2008

More terms from R. J. Mathar, May 23 2008

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A185783 Second accumulation array of A185780, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A027378 Expansion of (1+x^2-x^3)/(1-x)^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A139798 Coefficient of x^5 in (1-x-x^2)^(-n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions