A264399 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with even multiplicities.
1, 1, 1, 1, 3, 2, 3, 5, 2, 6, 4, 1, 9, 6, 9, 11, 2, 16, 13, 1, 20, 15, 7, 25, 28, 3, 32, 33, 11, 1, 40, 52, 9, 54, 55, 24, 2, 69, 82, 25, 84, 101, 40, 6, 101, 148, 46, 2, 136, 163, 73, 13, 156, 239, 89, 6, 202, 274, 127, 23, 1, 244, 364, 170, 14, 306, 437, 211, 46, 2
Offset: 0
Examples
T(6,1) = 4 because we have [4,1*,1], [3*,3], [2,1*,1,1,1], and [1*,1,1,1,1,1] (parts with even multiplicities are marked). Triangle starts: 1; 1; 1, 1; 3; 2, 3; 5, 2; 6, 4, 1; ...
Links
- Alois P. Heinz, Rows n = 0..1000, flattened
Programs
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Maple
g := product(1+x^j/(1-x^(2*j))+t*x^(2*j)/(1-x^(2*j)), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 28 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 28 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form # second Maple program: b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add( expand(`if`(j>0 and j::even, 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, Nov 25 2015 -
Mathematica
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[j>0 && EvenQ[ 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 25 2015, after Alois P. Heinz *) -
PARI
T(n) = { Vec(prod(k=1, n, (1+x^k-x^(2*k)+y*x^(2*k))/(1-x^(2*k)) + O(x*x^n))) } { my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ Andrew Howroyd, Dec 22 2017
Formula
G.f.: G(t,x) = Product_{j>=1} ((1 + x^j - x^(2j) + tx^(2j))/(1-x^(2j))).
Comments