cp's OEIS Frontend

A115995 Sum of the sizes of the Durfee squares of all partitions of n.

%I A115995 #52 Feb 16 2025 08:33:00
%S A115995 0,1,2,3,6,9,16,23,36,52,76,106,152,207,286,386,522,691,920,1202,1576,
%T A115995 2038,2636,3373,4320,5478,6944,8738,10984,13717,17116,21232,26308,
%U A115995 32441,39944,48977,59970,73147,89090,108151,131090,158417,191166,230049,276444
%N A115995 Sum of the sizes of the Durfee squares of all partitions of n.
%C A115995 Also sum of positive cranks of all partitions of n, n>1; see A064391. - _Vladeta Jovovic_, Oct 20 2006
%C A115995 This sequence, its author and the author of the above comment were mentioned in the Andrews-Chan-Kim paper, where it is called C_1 (see the remark on page 6). - _Omar E. Pol_, Apr 06 2012
%D A115995 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
%D A115995 G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
%H A115995 Alois P. Heinz, <a href="/A115995/b115995.txt">Table of n, a(n) for n = 0..3000</a>
%H A115995 George E. Andrews, <a href="http://www.math.psu.edu/andrews/pdf/80.pdf">Partitions and Durfee Dissection</a>
%H A115995 George E. Andrews, Song Heng Chan, and Byungchan Kim, <a href="http://www.math.psu.edu/andrews/pdf/292.pdf">The odd moments of ranks and cranks</a>
%H A115995 George E. Andrews, Frank G. Garvan, and Jie Liang, <a href="http://qseries.org/fgarvan/papers/spt-parity.pdf">Self-conjugate vector partitions and the parity of the spt-function</a>.
%H A115995 Atul Dixit, Bibekananda Maji, <a href="https://arxiv.org/abs/1806.04424">Partition implications of a new three parameter q-series identity</a>, arXiv:1806.04424 [math.CO], 2018.
%H A115995 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DurfeeSquare.html">Durfee Square.</a>
%F A115995 G.f.: Sum_{k>=1} (k*z^(k^2) / Product_{j=1..k} (1 - z^j)^2 ).
%F A115995 a(n) = Sum_{k=1..floor(sqrt(n))} k*A115994(n,k).
%F A115995 Convolution of A067742 and A000041. - _Vladeta Jovovic_, Oct 20 2006
%F A115995 a(n) = A195012(n) + A209616(n), n >= 1. - _Omar E. Pol_, Apr 06 2012
%F A115995 a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)). - _Vaclav Kotesovec_, Jan 02 2019
%e A115995 a(4) = 6 because the partitions [4], [3,1], [2,2], [2,1,1] and [1,1,1,1] of 4 have Durfee squares of sizes 1,1,2,1 and 1, respectively.
%p A115995 g:= add(k*z^(k^2)/mul((1-z^j)^2,j=1..k),k=1..10): gser:=series(g,z=0,56): seq(coeff(gser,z,n), n=0..52);
%p A115995 # second Maple program:
%p A115995 b:= proc(n, i) option remember;
%p A115995       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A115995     end:
%p A115995 a:= n-> add(add(b(k, d)*b(n-d^2-k, d), k=0..n-d^2)*d, d=1..isqrt(n)):
%p A115995 seq(a(n), n=0..70);  # _Alois P. Heinz_, Apr 09 2012
%p A115995 # Third Maple program, based on Theorem 1 of Andrews-Chan-Kim:
%p A115995 M:=101;
%p A115995 qinf:=mul(1-q^i,i=1..M);
%p A115995 qinf:=series(qinf,q,M);
%p A115995 C1:=add((-1)^(n+1)*q^(n*(n+1)/2)/(1-q^n),n=1..M);
%p A115995 C1:=series(C1/qinf,q,M);
%p A115995 seriestolist(%); # _N. J. A. Sloane_, Sep 04 2012
%t A115995 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]] ; a[n_] := Sum[ Sum[b[k, d]*b[n - d^2 - k, d], {k, 0, n - d^2}]*d, {d, 1, Sqrt[n]}]; Table [a[n], {n, 0, 70}] (* _Jean-François Alcover_, Jan 16 2015, after _Alois P. Heinz_ *)
%o A115995 (PARI) N=66; x='x+O('x^N); concat([0], Vec( sum(n=0,N, n*x^(n^2) / prod(k=1,n, 1-x^k)^2))) \\ _Joerg Arndt_, Mar 26 2014
%o A115995 (Sage)
%o A115995 [sum(p.frobenius_rank() for p in Partitions(n)) for n in range(45)] # _Peter Luschny_, Sep 15 2014
%Y A115995 Cf. A115994, A115720, A115721, A115722.
%K A115995 nonn
%O A115995 0,3
%A A115995 _Emeric Deutsch_, Feb 11 2006
%E A115995 Edited and verified by _Franklin T. Adams-Watters_, Mar 11 2006