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A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3. 13
1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 58, 66, 73, 82, 91, 102, 113, 126, 139, 155, 171, 190, 209, 232, 255, 282, 310, 342, 375, 413, 452, 497, 544, 596, 651, 713, 778, 850, 927, 1011, 1101, 1200, 1305, 1420, 1544, 1677, 1821, 1977, 2144, 2324, 2519, 2728 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Also number of partitions of n into distinct parts in which the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Aug 06 2004

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: sum(i>=1, x^(3*A000217(i)-2*i)/product(j=1..i, 1-x^j)). - Jon Perry, Jul 20 2004

G.f.: sum(n>=0, x^(n*(3*n-1)/2)/prod(k=1..n,1-x^k)). - Joerg Arndt, Jan 29 2011

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*r*sqrt(Pi*(1+3*r^2))), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the root of the equation r^3 + r = 1 and c = 3*(log(r))^2/2 + polylog(2, 1-r) = 0.566433354765746647188050807325058683443823543741343518... . - Vaclav Kotesovec, Jan 02 2016

EXAMPLE

a(12) = 6 because we have 12 = 11+1 = 10+2 = 9+3 = 8+4 = 7+4+1.

MATHEMATICA

nn=50; CoefficientList[Series[Sum[x^(j(3j-1)/2)Product[1/(1-x^i), {i, 1, j}], {j, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 21 2013 *)

CROSSREFS

Cf. A003114, A263719.

Column k=3 of A194543.

Sequence in context: A112672 A025765 A029029 * A006141 A185229 A026825

Adjacent sequences:  A025154 A025155 A025156 * A025158 A025159 A025160

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Prepended a(0)=1, Joerg Arndt, Jul 21 2013

STATUS

approved

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Last modified April 26 17:21 EDT 2017. Contains 285449 sequences.