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A107379
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Number of ways to write n^2 as the sum of n odd numbers, disregarding order.
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8
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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; internal format)
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OFFSET
| 0,4
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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. [From Paul D. Hanna, Feb 05 2012]
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..200
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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). [From Paul D. Hanna, Feb 05 2012]
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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.
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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 f(n-1, k-1) + f(n-2*k, k)
end proc;
seq(f(k^2, k), k=0..20);
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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 */
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CROSSREFS
| Cf. A152140.
Sequence in context: A091699 A129167 A151472 * A117428 A134168 A124427
Adjacent sequences: A107376 A107377 A107378 * A107380 A107381 A107382
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KEYWORD
| nonn,easy,changed
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AUTHOR
| David Radcliffe (dradcliffe(AT)gmail.com), Sep 25 2009
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EXTENSIONS
| Arguments in the Maple program swapped and 4 terms added by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 02 2009
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