A107379 Number of ways to write n^2 as the sum of n odd numbers, disregarding order.

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

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

Cf. A072243, A152140, A258191, A258192, A258234, A281489.

Sequence in context: A091699 A129167 A151472 * A117428 A339835 A134168

Adjacent sequences: A107376 A107377 A107378 * A107380 A107381 A107382

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