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A299730 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 ).

%I A299730 #31 Mar 08 2021 08:39:49
%S A299730 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 A299730 72,88,74,53,31,18,8,3,1,58,136,161,155,115,77,46,25,12,5,2,102,226,
%U A299730 307,291,241,168,110,65,35,18,8,3,1
%N A299730 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 A299730 Sequence of row lengths = A001651.
%H A299730 J. Stauduhar, <a href="/A299730/b299730.txt">Table of n, a(n) for n = 0..719</a>
%F A299730 T(n,k) = A222656(3n,k).
%e A299730 The irregular triangle T(n, k) begins:
%e A299730 3n\k  0   1   2   3   4   5   6   7   8   9
%e A299730 0:    1
%e A299730 3:    1   2
%e A299730 6:    3   4   3   1
%e A299730 9:    6   9   8   5   2
%e A299730 12:  12  20  19  14   8   3   1
%e A299730 15:  19  41  42  34  21  12   5   2
%e A299730 18:  37  72  88  74  53  31  18   8   3   1
%p A299730 b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, 1,
%p A299730       add(b(n-i*j, i-1)*`if`(isprime(i), x^j, 1), j=0..n/i)))
%p A299730     end:
%p A299730 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(3*n$2)):
%p A299730 seq(T(n), n=0..12);  # _Alois P. Heinz_, Mar 03 2018
%t A299730 b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, 1,
%t A299730      Sum[b[n - i*j, i - 1]*If[PrimeQ[i], x^j, 1], {j, 0, n/i}]]];
%t A299730 T[n_] := CoefficientList[b[3n, 3n], x];
%t A299730 T /@ Range[0, 12] // Flatten (* _Jean-François Alcover_, Mar 08 2021, after _Alois P. Heinz_ *)
%Y A299730 Cf. A001651, A008585, A222656.
%K A299730 nonn,tabf
%O A299730 0,3
%A A299730 _J. Stauduhar_, Feb 17 2018