%I #31 Mar 08 2021 08:39:49

%S 1,1,2,3,4,3,1,6,9,8,5,2,12,20,19,14,8,3,1,19,41,42,34,21,12,5,2,37,

%T 72,88,74,53,31,18,8,3,1,58,136,161,155,115,77,46,25,12,5,2,102,226,

%U 307,291,241,168,110,65,35,18,8,3,1

%N Irregular triangle read by rows: T(n,k) is the number of partitions of 3*n having exactly k prime parts; n >= 0, 0 <= k <= floor( 3*n / 2 ).

%C Sequence of row lengths = A001651.

%H J. Stauduhar, <a href="/A299730/b299730.txt">Table of n, a(n) for n = 0..719</a>

%F T(n,k) = A222656(3n,k).

%e The irregular triangle T(n, k) begins:

%e 3n\k 0 1 2 3 4 5 6 7 8 9

%e 0: 1

%e 3: 1 2

%e 6: 3 4 3 1

%e 9: 6 9 8 5 2

%e 12: 12 20 19 14 8 3 1

%e 15: 19 41 42 34 21 12 5 2

%e 18: 37 72 88 74 53 31 18 8 3 1

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

%p add(b(n-i*j, i-1)*`if`(isprime(i), x^j, 1), j=0..n/i)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(3*n$2)):

%p seq(T(n), n=0..12); # _Alois P. Heinz_, Mar 03 2018

%t b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, 1,

%t Sum[b[n - i*j, i - 1]*If[PrimeQ[i], x^j, 1], {j, 0, n/i}]]];

%t T[n_] := CoefficientList[b[3n, 3n], x];

%t T /@ Range[0, 12] // Flatten (* _Jean-François Alcover_, Mar 08 2021, after _Alois P. Heinz_ *)

%Y Cf. A001651, A008585, A222656.

%K nonn,tabf

%O 0,3

%A _J. Stauduhar_, Feb 17 2018