login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A242618 Number T(n,k) of partitions of n, where k is the difference between the number of odd parts and the number of even parts, both counted without multiplicity; triangle T(n,k), n>=0, read by rows. 28
1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 4, 2, 1, 1, 2, 3, 3, 2, 1, 8, 3, 3, 2, 4, 6, 5, 5, 4, 13, 8, 4, 1, 5, 5, 11, 13, 7, 1, 11, 20, 14, 9, 2, 1, 6, 13, 17, 26, 11, 3, 1, 22, 31, 27, 15, 5, 2, 12, 18, 34, 44, 18, 7, 4, 40, 47, 51, 23, 11, 5, 16, 36, 56, 72, 34, 11, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

T(n,0) = A241638(n).

Sum_{k<0} T(n,k) = A241640(n).

Sum_{k<=0} T(n,k) = A241639(n).

Sum_{k>=0} T(n,k) = A241637(n).

Sum_{k>0} T(n,k) = A241636(n).

T(n^2,n) = T(n^2+n,-n) = 1.

T(n^2+n,n) = Sum_{k} T(n,k) = A000041(n).

T(n^2+3*n,-n) = A000712(n).

LINKS

Alois P. Heinz, Rows n = 0..500, flattened

EXAMPLE

Triangle T(n,k) begins:

: n\k : -3  -2  -1   0   1   2   3 ...

+-----+---------------------------

:  0  :              1;

:  1  :                  1;

:  2  :          1,  0,  1;

:  3  :              1,  2;

:  4  :          2,  1,  1,  1;

:  5  :              4,  2,  1;

:  6  :      1,  2,  3,  3,  2;

:  7  :          1,  8,  3,  3;

:  8  :      2,  4,  6,  5,  5;

:  9  :          4, 13,  8,  4,  1;

: 10  :      5,  5, 11, 13,  7,  1;

: 11  :         11, 20, 14,  9,  2;

: 12  :  1,  6, 13, 17, 26, 11,  3;

: 13  :      1, 22, 31, 27, 15,  5;

: 14  :  2, 12, 18, 34, 44, 18,  7;

MAPLE

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

      expand(b(n, i-1)+add(b(n-i*j, i-1)*x^(2*irem(i, 2)-1), j=1..n/i))))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):

seq(T(n), n=0..20);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Expand[b[n, i - 1] + Sum[b[n - i*j, i - 1]*x^(2*Mod[i, 2] - 1), {j, 1, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-Fran├žois Alcover, Dec 12 2016 after Alois P. Heinz *)

CROSSREFS

Columns k=(-10)-10 give: A242682, A242683, A242684, A242685, A242686, A242687, A242688, A242689, A242690, A242691, A241638, A242692, A242693, A242694, A242695, A242696, A242697, A242698, A242699, A242700, A242701.

Row sums give A000041.

Cf. A240009 (parts counted with multiplicity), A240021 (distinct parts), A242626 (compositions counted without multiplicity).

Sequence in context: A259922 A162741 A104320 * A180264 A225200 A128706

Adjacent sequences:  A242615 A242616 A242617 * A242619 A242620 A242621

KEYWORD

nonn,tabf,look

AUTHOR

Alois P. Heinz, May 19 2014

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified April 27 01:03 EDT 2018. Contains 303149 sequences. (Running on oeis4.)