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%I #51 Sep 11 2017 15:48:28
%S 1,1,1,3,9,30,110,436,1801,7657,33401,148847,674585,3100410,14422567,
%T 67792847,321546251,1537241148,7400926549,35854579015,174677578889,
%U 855312650751,4207291811538,20782253017825,103048079556241,512753419159803,2559639388956793
%N Number of ways to write n^2 as the sum of n odd numbers, disregarding order.
%C Motivated by the fact that the n-th square is equal to the sum of the first n odd numbers.
%C Also the number of partitions of n^2 into n distinct parts. a(3) = 3: [1,2,6], [1,3,5], [2,3,4]. - _Alois P. Heinz_, Jan 20 2011
%C Also the number of partitions of n*(n-1)/2 into parts not greater than n. - _Paul D. Hanna_, Feb 05 2012
%C Also the number of partitions of n*(n+1)/2 into n parts. - _J. Stauduhar_, Sep 05 2017
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A107379/b107379.txt">Table of n, a(n) for n = 0..500</a> (first 200 terms from Alois P. Heinz)
%F a(n) = A008284((n^2+n)/2,n) = A008284(A000217(n),n). - _Max Alekseyev_, Sep 25 2009
%F a(n) = [x^(n*(n-1)/2)] Product_{k=1..n} 1/(1 - x^k). - _Paul D. Hanna_, Feb 05 2012
%F a(n) ~ c * d^n / n^2, where d = 5.400871904118154152466091119104270052029... = A258234, c = 0.155212227152682180502977404265024265... . - _Vaclav Kotesovec_, Sep 07 2014
%e For example, 9 can be written as a sum of three odd numbers in 3 ways: 1+1+7, 1+3+5 and 3+3+3.
%p f := proc (n, k) option remember;
%p if n = 0 and k = 0 then return 1 end if;
%p if n <= 0 or n < k then return 0 end if;
%p if `mod`(n+k, 2) = 1 then return 0 end if;
%p if k = 1 then return 1 end if;
%p return procname(n-1, k-1) + procname(n-2*k, k)
%p end proc;
%p seq(f(k^2,k), k=0..20);
%t Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n-1)/2}],{n,0,20}] (* _Vaclav Kotesovec_, May 25 2015 *)
%o (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n*(n-1)/2)))),n*(n-1)/2)} /* _Paul D. Hanna_ */
%Y Cf. A072243, A152140, A258191, A258192, A258234, A281489.
%K nonn,easy
%O 0,4
%A _David Radcliffe_, Sep 25 2009
%E Arguments in the Maple program swapped and 4 terms added by _R. J. Mathar_, Oct 02 2009
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