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A341719 Number of partitions of n into 9 primes (counting 1 as a prime).

%I A341719 #17 Feb 26 2022 04:23:39
%S A341719 1,1,2,2,4,4,7,7,11,11,16,15,23,21,30,27,39,35,51,44,63,54,78,67,97,
%T A341719 81,116,96,139,115,166,133,194,155,227,180,265,206,305,236,351,271,
%U A341719 403,305,460,346,522,391,592,438,668,489,751,551,844,608,942,674,1050,750
%N A341719 Number of partitions of n into 9 primes (counting 1 as a prime).
%p A341719 b:= proc(n, i) option remember; series(`if`(n=0, 1,
%p A341719      `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
%p A341719      `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 10)
%p A341719     end:
%p A341719 a:= n-> coeff(b(n, numtheory[pi](n)), x, 9):
%p A341719 seq(a(n), n=9..68);  # _Alois P. Heinz_, Feb 24 2021
%t A341719 b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
%t A341719      If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
%t A341719      If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 10}];
%t A341719 a[n_] := Coefficient[b[n, PrimePi[n]], x, 9];
%t A341719 Table[a[n], {n, 9, 68}] (* _Jean-François Alcover_, Feb 26 2022, after _Alois P. Heinz_ *)
%Y A341719 Cf. A008578, A034891, A259200, A341945, A341946, A341947, A341948, A341949, A341950, A341951, A341972.
%K A341719 nonn
%O A341719 9,3
%A A341719 _Ilya Gutkovskiy_, Feb 24 2021