A086602 -id:A086602 - cp's OEIS frontend

A110554 Column 11 of table A105552.

56, 285, 954, 2366, 4711, 7936, 11712, 15448, 18450, 20155, 20329, 19078, 16746, 13780, 10644, 7712, 5235, 3325, 1970, 1081, 544, 247, 99, 33, 8, 1

Offset: 1

Alford Arnold, Jul 30 2005

A105552 is constructed by summing values from ordered Gaussian polynomials, therefore the present begins with one value each from the following diagonal sequences and sums to A047970(11).

			a(6) = A107601(6) = 7936.

Cf. A047969, A047970, A110553.

Cf. A000124, A000330, A086602, A089574, A107600, A107601, A109125, A109126, A109820, A108538, A109821, A110553, A110624.

A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 2, 1, 10, 14, 12, 6, 1, 15, 30, 39, 39, 20, 1, 21, 55, 95, 138, 142, 71, 1, 28, 91, 195, 364, 548, 551, 270, 1, 36, 140, 357, 804, 1564, 2317, 2278, 1100

Offset: 1

Alford Arnold, Apr 16 2009

The first five diagonals are essentially A000012, A000217, A000330, A086602 and A159571.
From Alford Arnold, Apr 20 2009: (Start)
After the first two diagonals, each additional diagonal is computed using blocks of source partitions (defined in A053445).
The size of each block increases by powers of two; e.g. 22, 33 222, 44 332 333 2222; etc.
Each source partition can be associated with a specific sequence as illustrated in the below example using partition 332: grow the leftmost value to form 432 then append "1" to form 3321. in like manner, generate 532 4321 and 33211 from the previously formed cases. Note that the number of arrangements are 3, 6+12, and 6+24+30 respectively and that we now have three terms of A006011: 3 18 and 60.
Next we note that 6 39 138 364 804 ... A159571 resulted from summing term by term, the sequences associated with partitions 44 332 333 and 2222:
1...5..14...30...55
3..18..60..150..315
1...7..25...65..140
1...9..39..119..294
(End)

Cf. A047970 (row sums), A000012, A000217, A000330, A086602, A159571.

Cf. A053445. - Alford Arnold, Apr 20 2009

Submitted without a definition. - N. J. A. Sloane, Apr 18 2009

More terms from Alford Arnold, Oct 06 2009

A188667 Ordered (2,2)-selections from the multiset {1,1,2,2,3,3,...,n,n}.

Original entry on oeis.org

0, 0, 3, 21, 72, 180, 375, 693, 1176, 1872, 2835, 4125, 5808, 7956, 10647, 13965, 18000, 22848, 28611, 35397, 43320, 52500, 63063, 75141, 88872, 104400, 121875, 141453, 163296, 187572, 214455, 244125, 276768, 312576, 351747, 394485, 441000

Offset: 0

Thomas Wieder, Apr 07 2011

Number of ordered (2,2)-selections which can be taken from the first 2n elements of A008619, the positive integers repeated.  Order does count among subselections, e.g. [[1,1],[2,2]] and [[2,2],[1,1]] are different (2,2)-selections. Order does not count within a subselection, e.g. [1,3] is equivalent to [3,1].
Many thanks to Alois P. Heinz, Joerg Arndt, and Olivier Gérard for pointing out bugs in earlier versions of this sequence and for their comments!
The number of (not ordered) (2,2)-selections from natural numbers repeated = A008619 is equal to A086602 (observed by Alois P. Heinz).
The number of ordered (1,1)-selections from natural numbers repeated = A008619 is equal to the squares = A000290.
The number of ordered (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the Oblong numbers = A002378.
The number of ordered (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to A033487.
The number of (not ordered) (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the triangular numbers = A000217.
The number of (not ordered) (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the tritriangular numbers = A050534.
For n>0, the terms of this sequence are related to A014209 by a(n) = sum( i*A014209(i), i=0..n-1 ). [Bruno Berselli, Dec 20 2013]

			Example: For n=3 there are 21 ordered selections of the type (2,2):
[[1,1],[2,2]], [[1,2],[1,2]], [[2,2],[1,1]], [[1,2],[2,3]],
[[1,3],[2,2]], [[2,2],[1,3]], [[2,3],[1,2]], [[1,1],[2,3]],
[[1,2],[1,3]], [[1,3],[1,2]], [[2,3],[1,1]], [[1,1],[3,3]],
[[1,3],[1,3]], [[3,3],[1,1]], [[1,2],[3,3]], [[1,3],[2,3]],
[[2,3],[1,3]], [[3,3],[1,2]], [[2,2],[3,3]], [[2,3],[2,3]],
[[3,3],[2,2]].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Quang T. Bach, Roshil Paudyal, Jeffrey B. Remmel, A Fibonacci analogue of Stirling numbers, arXiv:1510.04310 [math.CO], 2015 (see p. 28).
T. Wieder, Generation of all possible multiselections from a multiset, Progress in Applied Mathematics, 2(1) (2011), 61-66, DOI:10.3968/j.pam.1925252820110201.010. - Thomas Wieder, Oct 15 2011

Cf. A014209.

Table[n*(n + 4)*(n - 1)^2/4, {n, 0, 100}] (* Vincenzo Librandi, Oct 18 2012 *)

a(n) = n*(n+4)*(n-1)^2/4.

G.f.: 3*x^2*(x^2-2*x-1) / (x-1)^5.

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A110554 Column 11 of table A105552.

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A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

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A188667 Ordered (2,2)-selections from the multiset {1,1,2,2,3,3,...,n,n}.

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