A307392 Number of partitions of n with at most one part in the interval [i*(i+1)/2, i+(i*(i+1)/2)] for all nonnegative integers i.
1, 1, 1, 1, 2, 3, 3, 3, 3, 4, 6, 9, 11, 12, 12, 12, 13, 15, 18, 22, 27, 34, 42, 50, 56, 60, 63, 66, 70, 76, 84, 94, 106, 120, 136, 154, 177, 206, 241, 279, 317, 352, 381, 404, 423, 442, 464, 492, 528, 574, 630, 694, 764, 839, 920, 1008, 1104, 1213, 1341, 1494, 1674, 1878
Offset: 0
Keywords
Examples
a(0)=1 by definition of the empty partition. a(10)=6 because 10=9+1=8+2=7+3=6+4=6+3+1 (for example, you cannot take 5+5 or 7+2+1 because of the definition of a(n)).
Programs
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Maple
f:= n-> 1+add(x^j, j=n*(n+1)/2..n*(n+3)/2): a:= n-> coeff(mul(f(k), k=1..ceil((sqrt(9+8*n)-3)/2)), x, n): seq(a(n), n=0..61);
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PARI
f(n, x) = (1+sum(j=n*(n+1)/2, n*(n+3)/2, x^j)); a(n) = polcoef(prod(k=1, ceil((sqrt(9+8*n)-3)/2), f(k, x)), n, x); \\ version 2.11.0 or newer; Michel Marcus, Apr 08 2019
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PARI
first(n) = v = Vecrev(Vec(a(n))); vector(n, i, v[i]) \\ using a(n) from above \\ David A. Corneth, Apr 08 2019
Formula
G.f.: Product_{n>=0} (1 + Sum_{k=(n*(n+1)/2)..(n*(n+3)/2)} x^k).
Comments