A389344 - OEIS

A389344

Irregular triangle read by rows: T(n,k) is the number of partitions of n into k distinct parts containing k as a part.

1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 4, 3, 0, 1, 4, 3, 1, 0, 1, 5, 4, 0, 0, 1, 5, 5, 1, 0, 1, 6, 7, 2, 0, 1, 6, 8, 3, 0, 1, 7, 10, 4, 0, 1, 7, 11, 5, 1, 0, 1, 8, 14, 6, 0, 0, 1, 8, 15, 9, 1

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OFFSET

1,20

COMMENTS

Empirical observations: Columns converge to A064428. Column 2*k of this triangle is column k of A344649.

LINKS

Table of n, a(n) for n=1..88.

FORMULA

G.f.: Sum_{i>0} ( t^i * q^(i*(i+1)/2) * (q)_{i-1} * Sum_{m=1..i} (q^(m*(m-1)) / ((q)_{i-m}*((q)_{m-1})^2) ) ) where (a)_k = Product_{i>=0..k-1} (1-a*q^i).

EXAMPLE

Triangle begins:

k=1 2 3 4 5

n=1 [1]

n=2 [0]

n=3 [0, 1]

n=4 [0, 0]

n=5 [0, 1]

n=6 [0, 1, 1]

n=7 [0, 1, 0]

n=8 [0, 1, 1]

n=9 [0, 1, 2]

n=10 [0, 1, 2, 1]

n=11 [0, 1, 2, 0]

n=12 [0, 1, 3, 1]

n=13 [0, 1, 3, 2]

n=14 [0, 1, 4, 3]

n=15 [0, 1, 4, 3, 1]

...

T(14,2) = 1 counts: (12,2).

T(14,3) = 4 counts: (6,5,3), (7,4,3), (9,3,2), (10,3,1).

T(14,4) = 3 counts: (5,4,3,2), (6,4,3,1), (7,4,2,1).

PROG

(PARI)

qs(a, q, n) = {prod(k=0, n-1, 1-a*q^k)}

b(n) = {floor(1/2+sqrt(2*n))}

C_qt(rowmax) = { my(N = rowmax+1, q='q+O('q^N), g = sum(i=1, N, t^i*q^(i*(i+1)/2)*qs(q, q, i-1)*sum(m=1, i, q^(m*(m-1))/( qs(q, q, i-m)*qs(q, q, m-1)^2)))); vector(N-1, n, my(r = concat(Vecrev(polcoeff(g, n)), [0])); vector(b(n+1)-1, i, if(#r<2, 0, r[i+1])))}

CROSSREFS

Cf. A000009, A064428, (A240855 row sums), A240861, A343941, A344649.

Sequence in context: A237049 A101662 A091064 * A373005 A275760 A293388

Adjacent sequences: A389341 A389342 A389343 * A389345 A389346 A389347

KEYWORD

nonn,tabf,easy

AUTHOR

John Tyler Rascoe, Oct 01 2025

STATUS

approved