A258192 -id:A258192 - cp's OEIS frontend

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

David Radcliffe, Sep 25 2009

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

			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.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (first 200 terms from Alois P. Heinz)

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

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);

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n-1)/2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

{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 */

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

Arguments in the Maple program swapped and 4 terms added by R. J. Mathar, Oct 02 2009

A258234 Decimal expansion of a constant related to A107379.

Original entry on oeis.org

5, 4, 0, 0, 8, 7, 1, 9, 0, 4, 1, 1, 8, 1, 5, 4, 1, 5, 2, 4, 6, 6, 0, 9, 1, 1, 1, 9, 1, 0, 4, 2, 7, 0, 0, 5, 2, 0, 2, 9, 4, 3, 7, 7, 1, 0, 1, 9, 1, 6, 7, 0, 5, 7, 0, 9, 3, 1, 7, 0, 6, 0, 1, 4, 4, 8, 4, 4, 8, 5, 1, 5, 9, 5, 0, 7, 5, 8, 1, 7, 7, 8, 9, 8, 8, 7, 4, 7, 9, 2, 0, 0, 0, 0, 6, 2, 0, 6, 2, 7, 7, 6, 7, 0, 0

Offset: 1

Vaclav Kotesovec, May 24 2015

Limit n->infinity (Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx)^(1/n) = Limit n->infinity (A258191(n)/A258192(n))^(1/n) = 1/A258234 = 0.18515528932235959464731321119795428527382236445907508398560553036... .

			5.4008719041181541524660911191042700520294...

StackExchange - Mathematica, No response to an infinite limit

Cf. A107379, A173519, A258191, A258192, A258268, A258788.

r^2/(r-1) /.FindRoot[-PolyLog[2, 1-r] == Log[r]^2, {r, E}, WorkingPrecision->117] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A107379(n)^(1/n).

Equals limit n->infinity A173519(n)^(1/n).

More terms from Vaclav Kotesovec, Jun 09 2015

A258191 Numerator of Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx.

Original entry on oeis.org

1, 11, 293, 487, 129952159, 13084761625, 8277192566411, 576489266167410341, 2154341459717480222819111, 1562388737113054944319018297, 4507453407946726622146977923716952747, 46170199302621715634236277404186409941, 20107953791404084220109855379873778475523352268948164789

Offset: 1

Vaclav Kotesovec, May 23 2015

nonn

Limit n->infinity (a(n)/A258192(n))^(1/n) = 0.185155...

The limit is equal to 0.1851552893223595946473132111979542852738... = 1/5.400871904118154152466091119104270052029... (see A258234). - Vaclav Kotesovec, May 24 2015

			Product_{k=1..n} x^k*(1-x^k)
n=1 x - x^2
n=2 x^3 - x^4 - x^5 + x^6
n=3 x^6 - x^7 - x^8 + x^10 + x^11 - x^12
Integral Product_{k=1..n} x^k*(1-x^k) dx
n=1 x^2/2 - x^3/3
n=2 x^4/4 - x^5/5 - x^6/6 + x^7/7
n=3 x^7/7 - x^8/8 - x^9/9 + x^11/11 + x^12/12 - x^13/13
For Integral_{x=0..1} set x=1
n=1 1/2 - 1/3 = 1/6, a(1)=1
n=2 1/4 - 1/5 - 1/6 + 1/7 = 11/420, a(2)=11
n=3 1/7 - 1/8 - 1/9 + 1/11 + 1/12 - 1/13 = 293/72072, a(3)=293

Vaclav Kotesovec, Table of n, a(n) for n = 1..50
StackExchange - Mathematica, No response to an infinite limit

Cf. A258192, A258229, A258230, A258234.

nmax=15; p=1; Table[p=Expand[p*x^n*(1-x^n)]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]], {n,1,nmax}] // Numerator

A258229 Numerator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.

Original entry on oeis.org

1, 5, 41, 188, 20777, 126661, 375407075, 4551271607, 2186878968457691, 405572061653677013, 579868609560670025014303, 756499881167742750802544581, 90137667815984749912207449629, 12095883009361301429642260272492831583, 83142433646555338064479023776802561123293

Offset: 1

Vaclav Kotesovec, May 24 2015

nonn

Limit n->infinity a(n) / A258230(n) = limit n->infinity Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx = 8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3)-1) = A258232 = 0.368412535931433652321316597327851...

			Product_{k=1..n} (1-x^k)
n=1 1 - x
n=2 1 - x - x^2 + x^3
n=3 1 - x - x^2 + x^4 + x^5 - x^6
Integral Product_{k=1..n} (1-x^k) dx
n=1 x - x^2/2
n=2 x - x^2/2 - x^3/3 + x^4/4
n=3 x - x^2/2 - x^3/3 + x^5/5 + x^6/6 - x^7/7
For Integral_{x=0..1} set x=1
n=1 1 - 1/2 = 1/2, a(1) = 1
n=2 1 - 1/2 - 1/3 + 1/4 = 5/12, a(2) = 5
n=3 1 - 1/2 - 1/3 + 1/5 + 1/6 - 1/7 = 41/105, a(3) = 41

Vaclav Kotesovec, Table of n, a(n) for n = 1..69
StackExchange - Mathematica, No response to an infinite limit

Cf. A258230, A258191, A258192, A258232.

nmax=15; p=1; Table[p=Expand[p*(1-x^n)]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]], {n,1,nmax}] // Numerator

A258230 Denominator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.

Original entry on oeis.org

2, 12, 105, 495, 55440, 340340, 1012647636, 12304749600, 5920545668637600, 1098951951860282520, 1572101004939647757775200, 2051717579526635495717258016, 244523633377266327241371614400, 32818916025992059215981780272862841200

Offset: 1

Vaclav Kotesovec, May 24 2015

nonn

Limit n->infinity A258229(n) / a(n) = limit n->infinity Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx = 8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3)-1) = A258232 = 0.368412535931433652321316597327851...

			Product_{k=1..n} (1-x^k)
n=1 1 - x
n=2 1 - x - x^2 + x^3
n=3 1 - x - x^2 + x^4 + x^5 - x^6
Integral Product_{k=1..n} (1-x^k) dx
n=1 x - x^2/2
n=2 x - x^2/2 - x^3/3 + x^4/4
n=3 x - x^2/2 - x^3/3 + x^5/5 + x^6/6 - x^7/7
For Integral_{x=0..1} set x=1
n=1 1 - 1/2 = 1/2, a(1) = 2
n=2 1 - 1/2 - 1/3 + 1/4 = 5/12, a(2) = 12
n=3 1 - 1/2 - 1/3 + 1/5 + 1/6 - 1/7 = 41/105, a(3) = 105

Vaclav Kotesovec, Table of n, a(n) for n = 1..69
StackExchange - Mathematica, No response to an infinite limit

Cf. A258229, A258191, A258192, A258232.

nmax=15; p=1; Table[p=Expand[p*(1-x^n)]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]], {n,1,nmax}] // Denominator

cp's OEIS Frontend

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

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A258234 Decimal expansion of a constant related to A107379.

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A258191 Numerator of Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx.

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A258229 Numerator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.

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Mathematica

A258230 Denominator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.

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Mathematica