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%I #36 Mar 30 2021 12:30:12
%S 0,0,1,0,1,1,1,0,1,1,1,2,0,0,1,2,1,3,1,3,1,1,4,1,0,0,1,5,1,1,6,2,0,0,
%T 0,1,7,2,0,1,9,2,1,10,3,1,12,3,1,0,0,0,1,14,3,1,1,17,4,1,0,0,0,0,1,19,
%U 5,1,1,23,5,1,1,26,6,2,0,1,30,7,2,0,0,0,0,0,1
%N T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.
%H Alois P. Heinz, <a href="/A333365/b333365.txt">Rows n = 0..1000, flattened</a>
%F T(n,pi(n)) = A010051(n) for n > 1.
%F T(p,pi(p)) = 1 if p is prime.
%F T(prime(k),k) = 1 for k >= 1.
%F Recursion: T(n,k) = Sum_{q=k..pi(n-p)} T(n-p, q) with p := prime(k) and T(n,k) = 0 if n < p, or 1 if n = p. - _David James Sycamore_, Mar 28 2020
%e In the A000607(11) = 6 partitions of 11 into prime parts, (11), 335, 227, 2225, 2333, 22223 the least parts are 11 = prime(5) (once), 3 = prime(2)(once), and 2 = prime(1) (four times), whereas 5 and 7 (prime(3) and prime(4)) do not occur. Thus row 11 is [4,1,0,0,1].
%e Triangle T(n,k) begins:
%e 0 ;
%e 0 ;
%e 1 ;
%e 0, 1 ;
%e 1 ;
%e 1, 0, 1 ;
%e 1, 1 ;
%e 2, 0, 0, 1 ;
%e 2, 1 ;
%e 3, 1 ;
%e 3, 1, 1 ;
%e 4, 1, 0, 0, 1 ;
%e 5, 1, 1 ;
%e 6, 2, 0, 0, 0, 1 ;
%e 7, 2, 0, 1 ;
%e 9, 2, 1 ;
%e 10, 3, 1 ;
%e 12, 3, 1, 0, 0, 0, 1 ;
%e 14, 3, 1, 1 ;
%e 17, 4, 1, 0, 0, 0, 0, 1 ;
%e 19, 5, 1, 1 ;
%e ...
%p b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
%p add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
%p end:
%p T:= proc(n) option remember; (p-> seq(`if`(isprime(i),
%p coeff(p, x, i), [][]), i=2..max(2,degree(p))))(b(n, 2, x))
%p end:
%p seq(T(n), n=0..23);
%t b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, With[{q = NextPrime[p]}, Sum[b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]]]];
%t T[n_] := If[n < 2, {0}, MapIndexed[If[PrimeQ[#2[[1]]], #1, Nothing]&, Rest @ CoefficientList[b[n, 2, x], x]]];
%t T /@ Range[0, 23] // Flatten (* _Jean-François Alcover_, Mar 30 2021, after _Alois P. Heinz_ *)
%Y Columns k=1-2 give: A000607(n-2) for n>1, A099773(n-3) for n>2.
%Y Row sums give A000607 for n>0.
%Y Length of n-th row is A000720(A331634(n)) for n>1.
%Y Indices of rows without 1's: A330433.
%Y Cf. A000040, A000720, A010051, A333129, A333238, A333259.
%K nonn,tabf
%O 0,12
%A _Alois P. Heinz_, Mar 16 2020
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