A128081 -id:A128081 - cp's OEIS frontend

A128080 Triangle, read by rows of n(n-1)+1 terms, of coefficients of q in the q-analog of the odd double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 3, 6, 9, 12, 14, 15, 14, 12, 9, 6, 3, 1, 1, 4, 10, 19, 31, 45, 60, 74, 86, 94, 97, 94, 86, 74, 60, 45, 31, 19, 10, 4, 1, 1, 5, 15, 34, 65, 110, 170, 244, 330, 424, 521, 614, 696, 760, 801, 815, 801, 760, 696, 614, 521, 424, 330, 244, 170

Offset: 0

Paul D. Hanna, Feb 14 2007

See A128084 for the triangle of coefficients of q in the q-analog of the even double factorials.

			Triangle begins:
  1;
  1;
  1,1,1;
  1,2,3,3,3,2,1;
  1,3,6,9,12,14,15,14,12,9,6,3,1;
  1,4,10,19,31,45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1;
  ...

Paul D. Hanna, Rows n=0..31 of triangle, in flattened form.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A001147 (row sums); A128081 (central terms), A128082 (diagonal), A128083 (row squared sums); A128084.

b:= proc(n) option remember; `if`(n=0, 1,
      simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
    end:
T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 22 2021

Catenate@Table[CoefficientList[Cancel@FunctionExpand[-q QPochhammer[1/q, q^2, n + 1]/(1 - q)^(n + 1)], q], {n, 0, 6}] (* Vladimir Reshetnikov, Sep 22 2021 *)
T[n_] := If[n == 0, {1}, Product[(1 - q^(2 j - 1))/(1 - q), {j, 1, n}] + O[q]^(n (n + 1)) // CoefficientList[#, q]&];
Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Sep 27 2022 *)

T(n,k)=if(k<0 || k>n*(n-1),0,if(n==0,1,polcoeff(prod(j=1,n,(1-q^(2*j-1))/(1-q)),k,q)))
for(n=0,8,for(k=0,n*(n-1),print1(T(n,k),", "));print(""))

The row sums are A001147, the odd double factorial numbers (2n-1)!!.

A128082 A diagonal of the triangle A128080 of coefficients of q in the q-analog of the odd double factorials: a(n) = A128080(n+1,n).

Original entry on oeis.org

1, 1, 3, 9, 31, 110, 400, 1477, 5516, 20775, 78762, 300179, 1148995, 4413877, 17007798, 65707390, 254430080, 987162527, 3836843836, 14936223511, 58226118626, 227271470103, 888117198666, 3474154716353, 13603246639501, 53310945927025, 209093495360796

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

			a(n) is the n-th term in the q-analog of odd double factorial (2n+1)!!, in which the coefficients of q (triangle A128080) begin:
   1;
  (1);
   1,(1),1;
   1,2,(3),3,3,2,1;
   1,3,6,(9),12,14,15,14,12,9,6,3,1;
   1,4,10,19,(31),45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1;
The terms enclosed in parenthesis are initial terms of this sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..500

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

b:= proc(n) option remember; `if`(n=0, 1,
      simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
    end:
a:= n-> coeff(b(n+1), q, n):
seq(a(n), n=0..28);  # Alois P. Heinz, Sep 22 2021

a[n_] := SeriesCoefficient[Product[(1-q^(2k-1))/(1-q), {k, 1, n+1}], {q, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 31 2021 *)

a(n)=if(n<1,0,polcoeff(prod(k=1,n,(1-q^(2*k-1))/(1-q)),n-1,q))

a(n+1) = A181971(2*n,n). - Reinhard Zumkeller, Jul 09 2012

a(n) ~ c * 2^(2*n) / sqrt(n), where c = QPochhammer(1/2, 1/4) / sqrt(Pi) = 0.236633772766964806372497000634617466975260409008748... - Vaclav Kotesovec, Feb 07 2023, updated Mar 17 2024

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))}

A128085 Central coefficients of q in the q-analog of the even double factorials: a(n) = [q^([n^2/2])] Product_{j=1..n} (1-q^(2j))/(1-q).

Original entry on oeis.org

1, 1, 2, 8, 46, 340, 3210, 36336, 484636, 7394458, 127707302, 2454109404, 52091631896, 1207854671388, 30431260261770, 826657521349952, 24114046688034516, 751085176539860458, 24899882719111953556

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

See A128081 for central coefficients of q in the q-analog of the odd double factorials. Also, A000140 is the central coefficients of q-factorials, giving the maximum number of permutations on n letters having the same number of inversions.

			a(n) is the central term of the q-analog of even double factorials,
in which the coefficients of q (triangle A128084) begin:
n=0: (1);
n=1: (1),1;
n=2: 1,2,(2),2,1;
n=3: 1,3,5,7,(8),8,7,5,3,1;
n=4: 1,4,9,16,24,32,39,44,(46),44,39,32,24,16,9,4,1;
n=5: 1,5,14,30,54,86,125,169,215,259,297,325,(340),340,325,297,...;...
The terms enclosed in parenthesis are initial terms of this sequence.

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

Cf. A000165 ((2n)!!); A128084 (triangle), A128086 (diagonal); A128081.

a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-q^(2*k))/(1-q)),n^2\2,q))

A350307 a(n) is the constant term in the expansion of Product_{j=1..n} (Sum_{k=-j..j} x^k)^n.

Original entry on oeis.org

1, 1, 37, 100683, 42935363305, 4440604747662968975, 161247684066768055445081543753, 2819198261291991623302749353791096334609249, 31233334332507494719367656927521237896029724037781845363309

Offset: 0

Seiichi Manyama, Dec 23 2021

nonn

a(n) is the coefficient of x^(n^2 * (n+1)/2) in Product_{j=0..n} (Sum_{k=0..2*j} x^k)^n.

Cf. A128081, A128083, A350282, A350305, A350308.

a[n_] := Coefficient[Series[Product[Sum[x^k, {k, -j, j}]^n, {j, 1, n}], {x, 0, 0}], x, 0]; Array[a, 9, 0] (* Amiram Eldar, Dec 24 2021 *)

a(n) = polcoef(prod(j=1, n, sum(k=-j, j, x^k))^n, 0);

a(n) = polcoef(prod(j=1, n, sum(k=0, 2*j, x^k))^n, n^2*(n+1)/2);

cp's OEIS Frontend

A128080 Triangle, read by rows of n(n-1)+1 terms, of coefficients of q in the q-analog of the odd double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

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A128082 A diagonal of the triangle A128080 of coefficients of q in the q-analog of the odd double factorials: a(n) = A128080(n+1,n).

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

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A128085 Central coefficients of q in the q-analog of the even double factorials: a(n) = [q^([n^2/2])] Product_{j=1..n} (1-q^(2j))/(1-q).

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A350307 a(n) is the constant term in the expansion of Product_{j=1..n} (Sum_{k=-j..j} x^k)^n.

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