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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 (list; graph; refs; listen; history; text; internal format)
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
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 */
CROSSREFS
Sequence in context: A091699 A129167 A151472 * A117428 A339835 A134168
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

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Last modified April 23 02:53 EDT 2024. Contains 371906 sequences. (Running on oeis4.)