login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k. 35
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

N. J. A. Sloane, Transforms

FORMULA

G.f. of column k: Product_{j>=1} (1 + x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019

EXAMPLE

A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).

Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).

Square array begins:

  1,  1,  1,   1,   1,   1,  ...

  0,  1,  2,   3,   4,   5,  ...

  0,  1,  3,   6,  10,  15,  ...

  0,  2,  6,  13,  24,  40,  ...

  0,  2,  9,  24,  51,  95,  ...

  0,  3, 14,  42, 100, 206,  ...

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

     (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))

    end:

A:= (n, k)-> b(n$2, k):

seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019

MATHEMATICA

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

CROSSREFS

Columns k=0-32 give: A000007, A000009, A022567-A022596.

Rows n=0-2 give: A000012, A001477, A000217.

Main diagonal gives A270913.

Antidiagonal sums give A299106.

Cf. A144064, A286352, A308680.

Sequence in context: A017827 A279778 A094266 * A291652 A071569 A261835

Adjacent sequences:  A286332 A286333 A286334 * A286336 A286337 A286338

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, May 07 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 23 19:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)