A395222 Number of partitions of n into 5 distinct parts containing the part 5.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 9, 11, 15, 17, 22, 25, 31, 36, 43, 49, 58, 65, 76, 85, 97, 108, 123, 135, 152, 167, 186, 203, 225, 244, 269, 291, 318, 343, 374, 401, 435, 466, 503, 537, 578, 615, 660, 701, 749, 794, 847, 895, 952, 1005, 1066, 1123, 1189

Offset

0,19

Comments

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.

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).

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).

Links

Table of n, a(n) for n=0..69.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Formula

G.f.: G_{5,5}(q) = q^15 * (1 + q^2*(1+q+q^2+q^3)/(1-q) + q^6*(1+q+2*q^2+q^3+q^4)/((1-q)*(1-q^2)) + q^12*(1+q+q^2+q^3)/((1-q)*(1-q^2)*(1-q^3)) + q^20/((1-q)*(1-q^2)*(1-q^3)*(1-q^4))).

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

a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 35.

a(n) = [n>14]*floor((n^3 - 42*n^2 + 672*n - 4176)/144 + (n mod 2)*(n-12)/16 + [n<16] + [n<21] + [n=25]), where "[.]" denote the Iverson brackets. - Hoang Xuan Thanh, May 16 2026

Prog

(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^(5*j)*prod(k=1, 5-j, q^k/(1-q^k)))))

Crossrefs

Cf. A394827, A396058.

Cf. A275216, A343941.

Sequence in context: A364698 A102571 A251241 * A064278 A220570 A341057

Adjacent sequences: A395219 A395220 A395221 * A395224 A395225 A395226

Keyword

nonn

Author

Seiichi Manyama, May 15 2026

Status

approved