OFFSET
1,2
COMMENTS
Sum_{k=0..n-1} T(n,k) = A000041(n); T(n,0) + T(n,1) = n for n > 1;
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
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Aug 09 2004
STATUS
approved