A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).
1, 1, 1, 1, 2, 1, 4, 1, 5, 2, 7, 4, 10, 5, 15, 6, 1, 21, 8, 1, 28, 13, 1, 37, 18, 1, 50, 25, 2, 67, 31, 3, 88, 42, 5, 115, 55, 6, 150, 73, 8, 193, 93, 11, 248, 122, 15, 317, 154, 19, 402, 200, 24, 1, 508, 253, 30, 1, 640, 320, 41, 1, 802, 399, 53, 1, 1002, 503, 69, 1
Offset: 0
Examples
The partition [1,1,2,3,3,3,3,4,4,4] has 2 parts i of multiplicity i-1: 2 and 4. T(5,1) = 2, counting [1,1,1,2] and [2,3]. T(8,2) = 1, counting [2,3,3]. Triangle starts: 1; 1; 1, 1; 2, 1; 4, 1; 5, 2; 7, 4; ...
Links
- Alois P. Heinz, Rows n = 0..1000, flattened
Crossrefs
Programs
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Maple
g := mul((t-1)*x^(i*(i+1))+1/(1-x^(i+1)), i = 1 .. 100)/(1-x): gser := simplify(series(g, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form # second Maple program: b:= proc(n, i) option remember; expand( `if`(n=0, 1, `if`(i<1, 0, add( `if`(i-1=j, x, 1)*b(n-i*j, i-1), j=0..n/i)))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)): seq(T(n), n=0..30); # Alois P. Heinz, Oct 10 2016 -
Mathematica
b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, x, 1]*b[n-i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 08 2016 after Alois P. Heinz *)
Formula
G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^(i+1))).
Comments