A097364 - OEIS

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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.

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

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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-François 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