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 A116685 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). 12
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Same as A097364 without the 0's. 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]. Row 1 has one term; row n (n>=2) has n-1 terms. Row sums yield the partition numbers (A000041). T(n,0)=A000005(n) (number of divisors of n). T(n,1)=A049820(n) (n minus number of divisors of n). T(n,2)=A008805(n-4) for n>=4. Sum(k*T(n,k),k=0..n-2)=A116686 LINKS Alois P. Heinz, Rows n = 1..142, flattened G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014. Bernard L. S. Lin, Saisai Zheng, k-regular partitions and overpartitions with bounded part differences, The Raman. J. 56 (2021) 685-695 FORMULA G.f.: sum(i>=1, x^i/(1-x^i)/prod(j=1..i-1, 1-t*x^j) ). EXAMPLE Triangle starts: 01: 1 02: 2 03: 2 1 04: 3 1 1 05: 2 3 1 1 06: 4 2 3 1 1 07: 2 5 3 3 1 1 08: 4 4 6 3 3 1 1 09: 3 6 6 7 3 3 1 1 10: 4 6 10 7 7 3 3 1 1 11: 2 9 10 12 8 7 3 3 1 1 12: 6 6 15 14 13 8 7 3 3 1 1 13: 2 11 15 20 16 14 8 7 3 3 1 1 14: 4 10 21 22 24 17 ... T(6,2)=3 because we have [4,1,1],[3,2,1] and [2,2,1,1]. 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-2) od; # yields sequence in triangular form MATHEMATICA 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 *) CROSSREFS Cf. A000041, A000005, A049820, A008805, A116686, A097364. 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: A236097 A239319 A236468 * A355522 A268190 A241150 Adjacent sequences: A116682 A116683 A116684 * A116686 A116687 A116688 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Feb 23 2006 STATUS approved

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Last modified July 22 16:55 EDT 2024. Contains 374540 sequences. (Running on oeis4.)