A286744 -id:A286744 - cp's OEIS frontend

A224898 G.f.: Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k).

1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 2, 2, 2, 2, 2, 1, 1, 0, -1, -2, -2, -4, -4, -5, -5, -6, -5, -6, -4, -4, -3, -2, 1, 1, 4, 5, 8, 9, 12, 12, 15, 15, 17, 16, 18, 15, 16, 13, 13, 8, 7, 1, 0, -7, -9, -17, -19, -27, -29, -37, -38, -46, -46, -53, -51, -57, -53, -57, -51, -53, -45, -45, -32, -31

Offset: 0

Paul D. Hanna, Jul 24 2013

Conjecture: a(n+1) = A286744(n) - A286745(n). - George Beck May 13 2017

			G.f.: A(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...
where
A(x) = 1 - x^2/(1-x) + x^6/((1-x)*(1-x^2)) - x^12/((1-x)*(1-x^2)*(1-x^3)) + x^20/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) - x^30/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Paul D. Hanna)

Cf. A227620, A227543, A039924, A005169, A286744, A286745, A003106.

a(n)=polcoeff(sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k,1+x*O(x^n))),n)
for(n=0, 80, print1(a(n), ", "))

A286745 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an odd number of parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 22, 25, 28, 31, 34, 39, 42, 47, 52, 58, 64, 72, 79, 89, 99, 111, 123, 139, 154, 173, 193, 216, 240, 269, 298, 333, 369, 410, 453, 503, 554, 613, 674, 743, 815, 897, 981, 1077, 1177, 1288, 1405, 1536, 1672, 1825, 1985, 2163, 2350, 2558, 2776, 3019, 3275, 3557, 3856, 4186, 4534, 4919

Offset: 0

George Beck, May 13 2017

nonn

			a(12) = 2 because of the partitions of 12, 12 and 6+4+2 are the only two that satisfy all three conditions.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A286744, A003106, A224898.

b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+2, 1-t)))
    end:
a:= n-> b(n, 2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 23 2017

Table[Length@ Select[ip@n, Min[-Differences@#] >= 2 && Min@# >= 2 && OddQ@Length@# &], {n, 20}]

a(n) ~ exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * sqrt(5*phi) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2020

cp's OEIS Frontend

A224898 G.f.: Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k).

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