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A116596 Number of partitions of n having exactly 1 part that appears exactly once. 1
1, 1, 1, 2, 4, 4, 8, 8, 12, 16, 23, 24, 40, 45, 59, 72, 99, 108, 153, 171, 224, 263, 341, 377, 504, 567, 711, 821, 1035, 1153, 1467, 1648, 2028, 2317, 2841, 3171, 3923, 4403, 5308, 6014, 7250, 8095, 9778, 10949, 13018, 14672, 17400, 19405, 23061, 25769, 30243 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Column 1 of A116595.

LINKS

Table of n, a(n) for n=1..51.

FORMULA

G.f.=sum(x^j*(1-x^j)/(1-x^j+x^(2j)), j=1..infinity)product((1-x^j+x^(2j))/(1-x^j), j=1..infinity).

G.f. for number of partitions of n having exactly 1 part that appears exactly m times is sum(x^(m*j)*(1-x^j)/(1-x^(m*j)+x^((m+1)*j)), j=1..infinity)*product((1-x^(m*j)+x^((m+1)*j))/(1-x^j), j=1..infinity). - Vladeta Jovovic, May 01 2006

EXAMPLE

a(5)=4 because we have [5],[3,1,1],[2,2,1] and [2,1,1,1] ([4,1],[3,2] and [1,1,1,1,1] do not qualify).

MAPLE

f:=sum(x^j*(1-x^j)/(1-x^j+x^(2*j)), j=1..75)*product((1-x^j+x^(2*j))/(1-x^j), j=1..75): fser:=series(f, x=0, 73): seq(coeff(fser, x^n), n=1..55);

MATHEMATICA

z = 30; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; m1[p_] := Min[Map[Length, Split[p]]]; Table[Count[IntegerPartitions[n], p_ /; u[p] == m1[p]], {n, 0, z}]  (* Clark Kimberling, Apr 23 2014 *)

CROSSREFS

Cf. A116595.

Sequence in context: A188112 A333194 A166632 * A248692 A048656 A107848

Adjacent sequences:  A116593 A116594 A116595 * A116597 A116598 A116599

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Feb 18 2006

STATUS

approved

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Last modified September 24 14:46 EDT 2020. Contains 337321 sequences. (Running on oeis4.)