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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A097364 Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k. 14
1, 2, 0, 2, 1, 0, 3, 1, 1, 0, 2, 3, 1, 1, 0, 4, 2, 3, 1, 1, 0, 2, 5, 3, 3, 1, 1, 0, 4, 4, 6, 3, 3, 1, 1, 0, 3, 6, 6, 7, 3, 3, 1, 1, 0, 4, 6, 10, 7, 7, 3, 3, 1, 1, 0, 2, 9, 10, 12, 8, 7, 3, 3, 1, 1, 0, 6, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 0, 2, 11, 15, 20, 16, 14, 8, 7, 3, 3, 1, 1, 0, 4, 10, 21, 22, 24, 17 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sum_{k=0..n-1} T(n,k) = A000041(n); T(n,0) + T(n,1) = n for n > 1;

T(n,0) = A000005(n); T(n,1) = A049820(n) for n > 1;

T(n,2) = floor((n-2)/2))*(floor((n-2)/2)) + 1)/2 = A000217(floor((n-2)/2))) = A008805(n-4) for n > 3.

Without the 0's (which are of no consequence for the triangle) this sequence is A116685. - Emeric Deutsch, Feb 23 2006

LINKS

Reinhard Zumkeller, Rows n = 1..60 of triangle, flattened

G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

FORMULA

G.f.: Sum_{i>=1} x^i/((1 - x^i)*Product_{j=1..i-1} (1 - t*x^j)). - Emeric Deutsch, Feb 23 2006

EXAMPLE

Triangle starts:

01:  1

02:  2  0

03:  2  1  0

04:  3  1  1  0

05:  2  3  1  1  0

06:  4  2  3  1  1  0

07:  2  5  3  3  1  1 0

08:  4  4  6  3  3  1 1 0

09:  3  6  6  7  3  3 1 1 0

10:  4  6 10  7  7  3 3 1 1 0

11:  2  9 10 12  8  7 3 3 1 1 0

12:  6  6 15 14 13  8 7 3 3 1 1 0

13:  2 11 15 20 16 14 8 7 3 3 1 1 0

14:  4 10 21 22 24 17 ...

- Joerg Arndt, Feb 22 2014

T(8,0)=4: 8=4+4=2+2+2+2=1+1+1+1+1+1+1+1,

T(8,1)=4: 3+3+2=2+2+2+1+1=2+2+1+1+1+1=2+1+1+1+1+1+1,

T(8,2)=6: 5+3=4+2+2=3+3+1+1=3+2+2+1=3+2+1+1+1=3+1+1+1+1+1,

T(8,3)=3: 4+3+1=4+2+1+1=4+1+1+1+1,

T(8,4)=3: 6+2=5+2+1=5+1+1+1,

T(8,5)=1: 6+1+1,

T(8,6)=1: 7+1,

T(8,7)=0;

Sum_{k=0..7} T(8,k) = 4+4+6+3+3+1+1+0 = 22 = A000041(8).

MAPLE

g:=sum(x^i/(1-x^i)/product(1-t*x^j, j=1..i-1), i=1..50): gser:=simplify(series(g, x=0, 18)): for n from 1 to 15 do P[n]:=coeff(gser, x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n], t, j), j=0..n-1) od;

# yields sequence in triangular form # Emeric Deutsch, Feb 23 2006

MATHEMATICA

rows = 14; max = rows+2; col[k0_ /; k0 > 0] := col[k0] = Sum[x^(2*k + k0) / Product[(1-x^(k+j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x]&; col[0] := Table[Switch[n, 1, 0, 2, 1, _, n - 1 - col[1][[n]]], {n, 1, Length[col[1]]}]; Table[col[k][[n+2]], {n, 0, rows-1 }, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Sep 10 2017, after Alois P. Heinz *)

PROG

(Haskell)

a097364 n k = length [qs | qs <- pss !! n, last qs - head qs == k] where

   pss = [] : map parts [1..] where

         parts x = [x] : [i : ps | i <- [1..x],

                                   ps <- pss !! (x - i), i <= head ps]

a097364_row n = map (a097364 n) [0..n-1]

a097364_tabl = map a097364_row [1..]

-- Reinhard Zumkeller, Feb 01 2013

CROSSREFS

Cf. A116685 (same sequence with zeros omitted).

Columns k=3..10 give A128508, A218567, A218568, A218569, A218570, A218571, A218572, A218573. T(2*n,n) = A117989(n). - Alois P. Heinz, Nov 02 2012

Sequence in context: A195050 A127371 A036849 * A254204 A321361 A319517

Adjacent sequences:  A097361 A097362 A097363 * A097365 A097366 A097367

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, Aug 09 2004

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 February 23 04:44 EST 2019. Contains 320411 sequences. (Running on oeis4.)