A396058 - OEIS

%I #32 May 16 2026 05:23:00

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,5,7,9,12,14,19,22,27,32,39,

%T 44,53,60,70,79,91,101,116,128,144,159,178,194,216,235,259,281,308,

%U 332,363,390,423,454,491,524,565,602,646,687,735,779,832,880

%N Number of partitions of n into 5 distinct parts containing the part 3.

%C Let b(n,k,m) be the number of partitions of n into k distinct parts containing the part m. Define G_{k,m}(q) = Sum_{n>=0} b(n,k,m) * q^n.

%C Then G_{k,m}(q) = q^(k*(k+1)/2) * Sum_{i=1..min(k,m)} q^((k-i+1)*(m-i)) * q_binomial(m-1,i-1) / Product_{j=1..k-i} (1-q^j).

%C Also, G_{k,m}(q) = Sum_{i=1..k} (-1)^(i-1) * q^(m*i) * Product_{j=1..k-i} q^j/(1-q^j).

%H Seiichi Manyama, <a href="/A396058/b396058.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-2,0,0,1,1,-1).

%F G.f.: G_{5,3}(q) = q^15 * (1/((1-q)*(1-q^2)) + q^4*(1+q)/((1-q)*(1-q^2)*(1-q^3)) + q^10/((1-q)*(1-q^2)*(1-q^3)*(1-q^4))).

%F G.f.: Sum_{j=1..5} (-1)^(j-1) * q^(3*j) * Product_{k=1..5-j} q^k/(1-q^k).

%F a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 25.

%o (PARI) my(N=70, q='q+O('q^N)); concat([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(j=1, 5, (-1)^(j-1)*q^(3*j)*prod(k=1, 5-j, q^k/(1-q^k)))))

%Y Cf. A275216, A394827, A396012.

%K nonn

%O 0,18

%A _Seiichi Manyama_, May 15 2026