A264399 - OEIS

%I #22 Jun 14 2023 17:29:26

%S 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,

%T 1,40,52,9,54,55,24,2,69,82,25,84,101,40,6,101,148,46,2,136,163,73,13,

%U 156,239,89,6,202,274,127,23,1,244,364,170,14,306,437,211,46,2

%N Triangle read by rows: T(n,k) is the number of partitions of n having k parts with even multiplicities.

%C T(n,0) = A055922(n).

%C Row sums give A000041. - _Omar E. Pol_, Nov 21 2015

%H Alois P. Heinz, <a href="/A264399/b264399.txt">Rows n = 0..1000, flattened</a>

%F G.f.: G(t,x) = Product_{j>=1} ((1 + x^j - x^(2j) + tx^(2j))/(1-x^(2j))).

%e 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).

%e Triangle starts:

%e   1;

%e   1;

%e   1, 1;

%e   3;

%e   2, 3;

%e   5, 2;

%e   6, 4, 1;

%e   ...

%p 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

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p       expand(`if`(j>0 and j::even, x, 1)*b(n-i*j, i-1)), j=0..n/i)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):

%p seq(T(n), n=0..30);  # _Alois P. Heinz_, Nov 25 2015

%t 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_ *)

%o (PARI)

%o 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))) }

%o { my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ _Andrew Howroyd_, Dec 22 2017

%Y Cf. A000041, A055922, A264398, A264400.

%K nonn,look,tabf

%O 0,5

%A _Emeric Deutsch_, Nov 21 2015