A258234 -id:A258234 - 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

A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

Original entry on oeis.org

1, 1, 2, 7, 23, 84, 331, 1367, 5812, 25331, 112804, 511045, 2348042, 10919414, 51313463, 243332340, 1163105227, 5598774334, 27119990519, 132107355553, 646793104859, 3181256110699, 15712610146876, 77903855239751, 387609232487489, 1934788962992123

Offset: 0

Reinhard Zumkeller, Feb 20 2010

nonn

a(n) is also the number of partitions of n^3 into n distinct parts <= n*(n+1). a(3) = 7: [4,11,12], [5,10,12], [6,9,12], [6,10,11], [7,8,12], [7,9,11], [8,9,10]. - Alois P. Heinz, Jan 25 2012

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..720 (terms 0..200 from Alois P. Heinz)

Cf. A066655, A097356, A258234.

Table[Length[IntegerPartitions[n(n + 1)/2, n]], {n, 10}] (* Alonso del Arte, Aug 12 2011 *)
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)=
{
    local(tr=n*(n+1)/2, x='x+O('x^(tr+3)), gf);
    gf = 1 / prod(k=1,n, 1-x^k); /* g.f. for partitions into parts <=n */
    return( polcoeff( truncate(gf), tr ) );
} /* Joerg Arndt, Aug 14 2011 */

a(n) = A026820(A000217(n),n).

a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - Vaclav Kotesovec, Sep 07 2014

More terms from D. S. McNeil, Aug 12 2011

A258788 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)).

Original entry on oeis.org

1, 1, 3, 12, 47, 192, 811, 3539, 15765, 71362, 327748, 1524081, 7161629, 33958506, 162312471, 781305581, 3784573140, 18435578714, 90261022638, 443956543235, 2192796266004, 10872208762458, 54095648185434, 270029668955605, 1351943521270155, 6787479872751732

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..494

Cf. A258234, A258789, A258797.

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n

Table[SeriesCoefficient[1/Product[x^k*(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}], {x, 0, n*(n+3)/2}], {n, 0, 30}]

a(n) ~ c * d^n / n^2, where d = A258234 = 5.40087190411815415246609111910427005202943771019167057093170601448448... = r^2/(r-1), where r is the root of the equation polylog(2, 1-r) + log(r)^2 = 0, c = 2.578341962163260914344332458898614289944... .

A258268 Decimal expansion of a constant related to A206226.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 25 2015

			9.153370192454122461948530292401354540073...

Cf. A206226, A206227, A206240, A258234, A107379, A173519, A238016, A258789.

r^3/(r-1) /.FindRoot[-PolyLog[2, 1-r] == 3*Log[r]^2/2, {r, E}, WorkingPrecision->120] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A206226(n)^(1/n).
Equals limit n->infinity A206227(n)^(1/n).
Equals limit n->infinity A206240(n)^(1/n).

More digits from Vaclav Kotesovec, Jun 10 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

A258192 Denominator of Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx.

Original entry on oeis.org

6, 420, 72072, 760760, 1266697832400, 783333734619744, 3002950101013562700, 1253414030788528596187200, 27809824888100301666382826331840, 118802724769051077369996224554510800, 2005396188718644499811084404372455793370133120

Offset: 1

Vaclav Kotesovec, May 23 2015

nonn

Limit n->infinity (A258191(n)/a(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)=6
n=2 1/4 - 1/5 - 1/6 + 1/7 = 11/420, a(2)=420
n=3 1/7 - 1/8 - 1/9 + 1/11 + 1/12 - 1/13 = 293/72072, a(3)=72072

Vaclav Kotesovec, Table of n, a(n) for n = 1..49
StackExchange - Mathematica, No response to an infinite limit

Cf. A258191, 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}] // Denominator

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A258788 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A258268 Decimal expansion of a constant related to A206226.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A258192 Denominator of Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica