A116685 - OEIS

%I #28 Mar 02 2022 08:55:15

%S 1,2,2,1,3,1,1,2,3,1,1,4,2,3,1,1,2,5,3,3,1,1,4,4,6,3,3,1,1,3,6,6,7,3,

%T 3,1,1,4,6,10,7,7,3,3,1,1,2,9,10,12,8,7,3,3,1,1,6,6,15,14,13,8,7,3,3,

%U 1,1,2,11,15,20,16,14,8,7,3,3,1,1,4,10,21,22,24,17,14,8,7,3,3,1,1,4,11,21

%N Triangle read by rows: T(n,k) is number of partitions of n that have k parts smaller than the largest part (n>=1, k>=0).

%C Same as A097364 without the 0's.

%C Also number of partitions of n such that the difference between the largest and smallest parts is k (see A097364). Example: T(6,2)=3 because we have [4,2],[3,2,1] and [3,1,1,1].

%C Row 1 has one term; row n (n>=2) has n-1 terms.

%C Row sums yield the partition numbers (A000041).

%C T(n,0)=A000005(n) (number of divisors of n).

%C T(n,1)=A049820(n) (n minus number of divisors of n).

%C T(n,2)=A008805(n-4) for n>=4.

%C Sum(k*T(n,k),k=0..n-2)=A116686

%H Alois P. Heinz, <a href="/A116685/b116685.txt">Rows n = 1..142, flattened</a>

%H G. E. Andrews, M. Beck and N. Robbins, <a href="https://arxiv.org/abs/1406.3374">Partitions with fixed differences between largest and smallest parts</a>, arXiv:1406.3374 [math.NT], 2014.

%H Bernard L. S. Lin, Saisai Zheng, <a href="https://doi.org/10.1007/s11139-020-00311-9">k-regular partitions and overpartitions with bounded part differences</a>, The Raman. J. 56 (2021) 685-695

%F G.f.: sum(i>=1, x^i/(1-x^i)/prod(j=1..i-1, 1-t*x^j) ).

%e Triangle starts:

%e 01:  1

%e 02:  2

%e 03:  2  1

%e 04:  3  1  1

%e 05:  2  3  1  1

%e 06:  4  2  3  1  1

%e 07:  2  5  3  3  1  1

%e 08:  4  4  6  3  3  1 1

%e 09:  3  6  6  7  3  3 1 1

%e 10:  4  6 10  7  7  3 3 1 1

%e 11:  2  9 10 12  8  7 3 3 1 1

%e 12:  6  6 15 14 13  8 7 3 3 1 1

%e 13:  2 11 15 20 16 14 8 7 3 3 1 1

%e 14:  4 10 21 22 24 17 ...

%e T(6,2)=3 because we have [4,1,1],[3,2,1] and [2,2,1,1].

%p 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-2) od;

%p # yields sequence in triangular form

%t rows = 15; 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]]}]; Join[{1}, Table[ col[k][[n+2]], {n, 0, rows-1}, {k, 0, n-1}] // Flatten] (* _Jean-François Alcover_, Sep 11 2017, after _Alois P. Heinz_ *)

%Y Cf. A000041, A000005, A049820, A008805, A116686, A097364.

%Y 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

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Feb 23 2006