A127728 -id:A127728 - cp's OEIS frontend

A047653 Constant term in expansion of (1/2) * Product_{k=-n..n} (1 + x^k).

1, 2, 4, 10, 26, 76, 236, 760, 2522, 8556, 29504, 103130, 364548, 1300820, 4679472, 16952162, 61790442, 226451036, 833918840, 3084255128, 11451630044, 42669225172, 159497648600, 597950875256, 2247724108772, 8470205600640, 31991616634296, 121086752349064

Offset: 0

N. J. A. Sloane

nonn

Or, constant term in expansion of Product_{k=1..n} (x^k + 1/x^k)^2. - N. J. A. Sloane, Jul 09 2008
Or, maximal coefficient of the polynomial (1+x)^2 * (1+x^2)^2 *...* (1+x^n)^2.
a(n) = A000302(n) - A181765(n).
From Gus Wiseman, Apr 18 2023: (Start)
Also the number of subsets of {1..2n} that are empty or have mean n. The a(0) = 1 through a(3) = 10 subsets are:
  {}  {}   {}       {}
      {1}  {2}      {3}
           {1,3}    {1,5}
           {1,2,3}  {2,4}
                    {1,2,6}
                    {1,3,5}
                    {2,3,4}
                    {1,2,3,6}
                    {1,2,4,5}
                    {1,2,3,4,5}
Also the number of subsets of {-n..n} with no 0's but with sum 0. The a(0) = 1 through a(3) = 10 subsets are:
  {}  {}      {}           {}
      {-1,1}  {-1,1}       {-1,1}
              {-2,2}       {-2,2}
              {-2,-1,1,2}  {-3,3}
                           {-3,1,2}
                           {-2,-1,3}
                           {-2,-1,1,2}
                           {-3,-1,1,3}
                           {-3,-2,2,3}
                           {-3,-2,-1,1,2,3}
(End)

T. D. Noe, Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..1669 (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz)
Ovidiu Bagdasar and Dorin Andrica, New results and conjectures on 2-partitions of multisets, 2017 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO).
Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
R. C. Entringer, Representation of m as Sum_{k=-n..n} epsilon_k k, Canad. Math. Bull., 11 (1968), 289-293.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
R. P. Stanley, Weyl groups, the hard Lefschetz theorem and the Sperner property, SIAM J. Algebraic and Discrete Methods 1 (1980), 168-184.

Cf. A025591.
Cf. A053632; variant: A127728.
For median instead of mean we have A079309(n) + 1.
Odd bisection of A133406.
A000980 counts nonempty subsets of {1..2n-1} with mean n.
A007318 counts subsets by length, A327481 by mean.
Cf. A024718, A047997, A057552, A070925, A212352, A326512, A326513, A327475, A361866, A362046.

f:=n->coeff( expand( mul((x^k+1/x^k)^2,k=1..n) ),x,0);
# second Maple program:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(i=0, 1, 2*b(n, i-1)+b(n+i, i-1)+b(abs(n-i), i-1)))
    end:
a:=n-> b(0, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 10 2014

b[n_, i_] := b[n, i] = If[n>i*(i+1)/2, 0, If[i == 0, 1, 2*b[n, i-1]+b[n+i, i-1]+b[Abs[n-i], i-1]]]; a[n_] := b[0, n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 26; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2];
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n] +
    2 PadLeft[PadRight[d, Length[d] + n], Length[d] + 2 n];
, {n, nmax}];
a1 (* Ray Chandler, Mar 15 2014 *)
Table[Length[Select[Subsets[Range[2n]],Length[#]==0||Mean[#]==n&]],{n,0,6}] (* Gus Wiseman, Apr 18 2023 *)

PARI
```
a(n)=polcoeff(prod(k=-n,n,1+x^k),0)/2
```

{a(n)=sum(k=0,n*(n+1)/2,polcoeff(prod(m=1,n,1+x^m+x*O(x^k)),k)^2)} \\ Paul D. Hanna, Nov 30 2010

Sum of squares of coefficients in Product_{k=1..n} (1+x^k):
a(n) = Sum_{k=0..n(n+1)/2} A053632(n,k)^2. - Paul D. Hanna, Nov 30 2010
a(n) = A000980(n)/2.
a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 11 2014
From Gus Wiseman, Apr 18 2023 (Start)
a(n) = A133406(2n+1).
a(n) = A212352(n) + 1.
a(n) = A362046(2n) + 1.
(End)

More terms from Michael Somos, Jun 10 2000

A128083 Sum of squared coefficients of q in the q-analog of the odd double factorials.

Original entry on oeis.org

1, 1, 3, 37, 1159, 66953, 6158021, 825889193, 152147002939, 36866098462221, 11368538145120143, 4347671960639941039, 2019396728684584627337, 1119792551093682455434255, 730724550040451849614251167

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

Cf. A001147 ((2n-1)!!); A128080 (triangle), A128081 (central terms), A128082 (diagonal); A127728, A128087.

{a(n)=if(n==0,1,sum(k=0,n^2,polcoeff(prod(j=1,n,(1-q^(2*j-1))/(1-q)),k,q)^2))}

A129276 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk-k) in the squared q-factorial of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 8, 1, 1, 42, 106, 42, 1, 1, 241, 1558, 1558, 241, 1, 1, 1444, 23589, 53612, 23589, 1444, 1, 1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1, 1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1

Offset: 0

Paul D. Hanna, Apr 07 2007

Dual triangle is A129274.

Central terms form a bisection of A127728.

			Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 4 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, (42), 71, 96, (106), 96, 71, (42), 19, 6, (1)],
where the terms enclosed in parenthesis form row 4.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 8, 8, 1;
1, 42, 106, 42, 1;
1, 241, 1558, 1558, 241, 1;
1, 1444, 23589, 53612, 23589, 1444, 1;
1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1;
1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1; ...

Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A129277 (column 1), A129278 (column 2); A127728 (central terms), related triangles: A129274, A128564, A008302 (Mahonian numbers).

faq[n_, q_] := Product[(1-q^k)/(1-q), {k, 1, n}]; t[0, 0] = t[1, 0] = t[1, 1] = 1; t[n_, k_] := SeriesCoefficient[faq[n, q]^2, {q, 0, (n-1)*k}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 26 2013 *)

T(n,k)=if(n==0,1,polcoeff(prod(i=1,n,(1-x^i)/(1-x))^2,(n-1)*k))

T(n,k) = [q^(nk-k)] Product_{i=1..n} { (1-q^i)/(1-q) }^2 for n>0, with T(0,0)=1.

Row sums = (n!)^2/(n-1) for n>=2.

A128087 Sum of squared coefficients of q in the q-analog of the even double factorials.

Original entry on oeis.org

1, 2, 14, 296, 12938, 956720, 107245250, 16966970200, 3601980861720, 988252809411908, 340375635448973106, 143798619953044471444, 73123320014581106403732, 44060303354020797873285800

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

See A128083 for sum of squared coefficients of q in the q-analog of the odd double factorials. Also, A127728 is the sum of squared coefficients of q in the q-factorials.

Cf. A000165 ((2n)!!); A128084 (triangle), A128085 (central terms), A128086 (diagonal); A127728.

{a(n)=if(n==0,1,sum(k=0,n^2,polcoeff(prod(j=1,n,(1-q^(2*j))/(1-q)),k,q)^2))}

A129274 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 71, 71, 1, 1, 474, 1930, 474, 1, 1, 3103, 40096, 40096, 3103, 1, 1, 20190, 739929, 2108560, 739929, 20190, 1, 1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1, 1, 853176, 215022825, 3286786158, 7625997280

Offset: 0

Paul D. Hanna, Apr 07 2007

Row sums equal A010790(n) = n!*(n+1)! for n>=0. Central terms form a bisection of A127728. Dual triangle is A129276.

			Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 3 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, 42, (71), 96, 106, 96, (71), 42, 19, 6, (1)],
where the terms enclosed in parenthesis form row 3.
Triangle begins:
1;
1, 1;
1, 10, 1;
1, 71, 71, 1;
1, 474, 1930, 474, 1;
1, 3103, 40096, 40096, 3103, 1;
1, 20190, 739929, 2108560, 739929, 20190, 1;
1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1;
1, 853176, 215022825, 3286786158, 7625997280, 3286786158, 215022825, 853176, 1; ...

Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A129275 (column 1); A127728 (central terms), A010790 (row sums); related triangles: A129276, A128564, A008302 (Mahonian numbers).

T(n,k)=polcoeff(prod(i=1,n+1,(1-x^i)/(1-x))^2,(n+1)*k)

T(n,k) = [q^(nk+k)] Product_{i=1..n+1} { (1-q^i)/(1-q) }^2.

A380274 Sum of cubes of coefficients of q in the q-factorials.

Original entry on oeis.org

1, 1, 2, 18, 522, 34986, 4524240, 1003172616, 351349509504, 182985164256000, 135303274820730372, 136936922140937021688, 184146557651652262521738, 321051865325352021467189658, 710866983641078174204266934736, 1964068265459581480020247325821224

Offset: 0

Vaclav Kotesovec, Jan 18 2025

nonn

			a(4) = 1^3 + 3^3 + 5^3 + 6^3 + 5^3 + 3^3 + 1^3 = 522.

Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A008302, A127728, A380275.

Table[Total[CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n-1}]], x]^3], {n, 0, 15}]

a(n) = my(v=Vec(prod(k=1, n, (1-q^k)/(1-q)))); sum(i=1, #v, v[i]^3); \\ Michel Marcus, Jan 18 2025

a(n) = Sum_{k>=0} A008302(n,k)^3.

Conjecture: a(n) ~ 6*sqrt(3) * n!^3 / (Pi * n^3).

A380275 Sum of the fourth powers of the coefficients of q in the q-factorials.

Original entry on oeis.org

1, 1, 2, 34, 2710, 669142, 403186412, 504370709488, 1170803949124848, 4644277674894466168, 29557755573424568318844, 287158619888775996039794756, 4090368591132420991019182924018, 82628355729998755756059701468470738, 2301817961412922763844330401786521588244

Offset: 0

Vaclav Kotesovec, Jan 18 2025

nonn

Conjecture: In general, sum of the k-th powers of the coefficients of q in the q-factorials is asymptotic to 2^((k-1)/2) * 3^(k-1) * n!^k / (sqrt(k) * Pi^((k-1)/2) * n^(3*(k-1)/2)).

			a(4) = 1^4 + 3^4 + 5^4 + 6^4 + 5^4 + 3^4 + 1^4 = 2710.

Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A008302, A127728, A380274.

Table[Total[CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n-1}]], x]^4], {n, 0, 15}]

a(n) = my(v=Vec(prod(k=1, n, (1-q^k)/(1-q)))); sum(i=1, #v, v[i]^4); \\ Michel Marcus, Jan 18 2025

a(n) = Sum_{k>=0} A008302(n,k)^4.

Conjecture: a(n) ~ 27*sqrt(2) * n!^4 / (Pi^(3/2) * n^(9/2)).

cp's OEIS Frontend

A047653 Constant term in expansion of (1/2) * Product_{k=-n..n} (1 + x^k).

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A128083 Sum of squared coefficients of q in the q-analog of the odd double factorials.

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A129276 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk-k) in the squared q-factorial of n.

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A128087 Sum of squared coefficients of q in the q-analog of the even double factorials.

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A129274 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.

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A380274 Sum of cubes of coefficients of q in the q-factorials.

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A380275 Sum of the fourth powers of the coefficients of q in the q-factorials.

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