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A107379
Number of ways to write n^2 as the sum of n odd numbers, disregarding order.
14
1, 1, 1, 3, 9, 30, 110, 436, 1801, 7657, 33401, 148847, 674585, 3100410, 14422567, 67792847, 321546251, 1537241148, 7400926549, 35854579015, 174677578889, 855312650751, 4207291811538, 20782253017825, 103048079556241, 512753419159803, 2559639388956793
OFFSET
0,4
COMMENTS
Motivated by the fact that the n-th square is equal to the sum of the first n odd numbers.
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
Also the number of partitions of n*(n-1)/2 into parts not greater than n. - Paul D. Hanna, Feb 05 2012
Also the number of partitions of n*(n+1)/2 into n parts. - J. Stauduhar, Sep 05 2017
Also the number of fair dice with n sides and expected value (n+1)/2 with distinct composition of positive integers. - Felix Huber, Aug 11 2024
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (first 200 terms from Alois P. Heinz)
FORMULA
a(n) = A008284((n^2+n)/2,n) = A008284(A000217(n),n). - Max Alekseyev, Sep 25 2009
a(n) = [x^(n*(n-1)/2)] Product_{k=1..n} 1/(1 - x^k). - Paul D. Hanna, Feb 05 2012
a(n) ~ c * d^n / n^2, where d = 5.400871904118154152466091119104270052029... = A258234, c = 0.155212227152682180502977404265024265... . - Vaclav Kotesovec, Sep 07 2014
EXAMPLE
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.
MAPLE
f := proc (n, k) option remember;
if n = 0 and k = 0 then return 1 end if;
if n <= 0 or n < k then return 0 end if;
if `mod`(n+k, 2) = 1 then return 0 end if;
if k = 1 then return 1 end if;
return procname(n-1, k-1) + procname(n-2*k, k)
end proc;
seq(f(k^2, k), k=0..20);
MATHEMATICA
Table[SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}], {x, 0, n*(n-1)/2}], {n, 0, 20}] (* Vaclav Kotesovec, May 25 2015 *)
PROG
(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, Feb 05 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David Radcliffe, Sep 25 2009
EXTENSIONS
Arguments in the Maple program swapped and 4 terms added by R. J. Mathar, Oct 02 2009
STATUS
approved