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!)
A284593 Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common. 5
1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 2, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 2, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 6, 5, 6, 5, 6, 4, 5, 10, 12, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 12, 15, 7, 6, 8, 7, 8, 8, 8, 7, 8, 6, 7, 15 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Compare with A284592.

LINKS

Alois P. Heinz, Antidiagonals n = 0..200, flattened

H. S. Wilf, Lectures on Integer Partitions

FORMULA

O.g.f. Product_{j >= 1} (1 + x^j + y^j) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A032302.

EXAMPLE

Square array begins

  n\k| 0  1  2  3  4  5  6   7   8   9  10  11  12  13

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

  0  | 1  1  1  2  2  3  4   5   6   8  10  12  15  18: A000009

  1  | 1  0  1  1  1  2  2   3   3   5   5   7   8  10: A096765

  2  | 1  1  0  1  2  2  2   3   4   5   6   7   9  11: A015744

  3  | 2  1  1  2  2  3  4   6   6   8   9  12  15  18

  4  | 2  1  2  2  2  3  5   5   7   9  10  14  15  19

  5  | 3  2  2  3  3  6  6   8   9  12  16  19  22  28

  6  | 4  2  2  4  5  6  8   9  11  16  18  22  27  33

  7  | 5  3  3  6  5  8  9  14  16  20  23  29  34  41

  ...

T(3,7) = 6: the six pairs of partitions of 3 and 7 into distinct parts and with no parts in common are (3, 7), (3, 6 + 1), (3, 5 + 2), (3, 4 + 2 + 1), (2 + 1, 7) and (2 + 1, 4 + 3).

MAPLE

# A284593 as a square array

ser := taylor(taylor(mul(1 + x^j + y^j, j = 1..10), x, 11), y, 11):

convert(ser, polynom):

s := convert(%, polynom):

with(PolynomialTools):

for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;

# second Maple program:

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

      b(n, i-1)+expand((x^i+1)*b(n-i, min(n-i, i-1)))))

    end:

T:= (n, k)-> coeff(b(n+k$2), x, k):

seq(seq(T(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Aug 24 2019

MATHEMATICA

nmax = 12; M = CoefficientList[#, y][[;; nmax+1]]& /@ (Product[1 + x^j + y^j, {j, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& // Expand);

T[n_, k_] := M[[n+1, k+1]];

Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Dec 07 2019 *)

CROSSREFS

Cf. A000009 (row 0), A096765 (row 1), A015744 (row 2), A032032 (antidiagonal sums).

Cf. A284592, A322210.

Main diagonal gives 2*A108796 (for n>0).

Sequence in context: A171533 A115236 A307777 * A190672 A327910 A242998

Adjacent sequences:  A284590 A284591 A284592 * A284594 A284595 A284596

KEYWORD

nonn,tabl,easy

AUTHOR

Peter Bala, Mar 30 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 April 13 18:33 EDT 2021. Contains 342939 sequences. (Running on oeis4.)