A116860 Triangle read by rows: T(n,k) is the number of partitions into distinct odd parts with smallest part k (n>=1, k>=1).

1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Emeric Deutsch, Feb 27 2006

Keywords

• nonn
• tabf

Comments

Row 2 has no terms, row 2n-1 has 2n-1 terms, row 4n has 2n-1 terms, row 4n+2 for n>=1 has 2n-1 terms. Row sums are A000700. T(n,1) = A027349(n). Sum_{k>=1} k*T(n,k) = A092319(n).

Within those rows, T(n, k) = 0 occurs with an odd k iff n is odd and 2*floor((n+3)/6) - 1 <= k < n. - Álvar Ibeas, Aug 03 2020

Examples

			T(25,3) = 3 because we have [17,5,3], [15,7,3] and [13,9,3].
Triangle starts:
1;
{};
0,0,1;
1;
0,0,0,0,1;
1;
0,0,0,0,0,0,1;
1,0,1;
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened

Crossrefs

Cf. A000700, A027349, A092319, A116857.

Programs

Maple

g:=sum(t^(2*j-1)*x^(2*j-1)*product(1+x^(2*i-1),i=j+1..30),j=1..30): gser:=simplify(series(g,x=0,52)): for n from 1 to 19 do P[n]:=sort(coeff(gser,x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n=2 then 0 elif n mod 4 = 0 then n/2-1 else n/2-2 fi end: 1; {}; for n from 3 to 19 do seq(coeff(P[n],t^j),j=1..d(n)) od; # yields sequence in triangular form

Mathematica

imax = 20;
s = Sum[t^(2 j - 1)*x^(2 j - 1)*Product[1 + x^(2 i - 1), {i, j + 1, imax}], {j, 1, imax}] + O[x]^imax;
Rest /@ DeleteCases[CoefficientList[#, t]& /@ CoefficientList[s, x], {}] // Flatten (* Jean-François Alcover, May 22 2018 *)

Formula

G.f.: Sum_(t^(2*j-1)*x^(2*j-1)*Product_(1+x^(2*i-1), i=j+1..infinity), j=1..infinity).

For k even, T(n, k) = 0. For k odd, T(n, n) = 1 and, if k < n, T(n, k) = Sum_{i > k} T(n - k, i). - Álvar Ibeas, Aug 03 2020