A128080 -id:A128080 - cp's OEIS frontend

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).

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

A128084 Triangle, read by rows of n^2+1 terms, of coefficients of q in the q-analog of the even double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 5, 7, 8, 8, 7, 5, 3, 1, 1, 4, 9, 16, 24, 32, 39, 44, 46, 44, 39, 32, 24, 16, 9, 4, 1, 1, 5, 14, 30, 54, 86, 125, 169, 215, 259, 297, 325, 340, 340, 325, 297, 259, 215, 169, 125, 86, 54, 30, 14, 5, 1, 1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919

Offset: 0

Paul D. Hanna, Feb 14 2007

See A128080 for the triangle of coefficients of q in the q-analog of the odd double factorials.

Row maxima ~ 2^n*n!/(sigma * sqrt(2*Pi)), sigma^2 = (4*n^3 + 6*n^2 - n)/36 = variance of Coxeter group B_n (see also A161858). - Mikhail Gaichenkov, Feb 08 2023

			The row sums form A000165, the even double factorial numbers:
  [1, 2, 8, 48, 384, 3840, 46080, 645120, ..., (2n)!!, ...].
Triangle begins:
 1;
 1, 1;
 1, 2,  2,  2,  1;
 1, 3,  5,  7,  8,  8,   7,   5,   3,   1;
 1, 4,  9, 16, 24, 32,  39,  44,  46,  44,  39,  32,  24,  16,   9,   4,   1;
 ...

Paul D. Hanna, Rows n=0..30 of triangle, in flattened form.
Hasan Arslan, A combinatorial interpretation of Mahonian numbers of type B, arXiv:2404.05099 [math.CO], 2024.
Thomas Kahle and Christian Stump, Counting inversions and descents of random elements in finite Coxeter groups, arXiv:1802.01389 [math.CO], 2018-2019.
Ali Kessouri, Moussa Ahmia, Hasan Arslan, and Salim Mesbahi, Combinatorics of q-Mahonian numbers of type B and log-concavity, arXiv:2408.02424 [math.CO], 2024. See p. 6.
Eric Weisstein's World of Mathematics, q-Factorial.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
A. V. Yurkin, New binomial and new view on light theory, LAP Lambert Academic Publishing, 2013, 78 pages.

Cf. A000165 ((2n)!!); A128085 (central terms); A128086 (diagonal), A128087 (row squared sums); A128080, A002522 (row lengths).

The growth series for the affine Coxeter groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858.

t[n_, k_] := If[k < 0 || k > n^2, 0, If[n == 0, 1, Coefficient[ Series[ Product[ (1 - q^(2*j))/(1 - q), {j, 1, n}], {q, 0, n^2}], q, k]]]; Table[t[n, k], {n, 0, 6}, {k, 0, n^2}] // Flatten (* Jean-François Alcover, Mar 06 2013, translated from Pari *)

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

row(n)=Vec(prod(j=1, n, (1-x^(2*j))/(1-x))) \\ Andrew Howroyd, Mar 21 2025

A128081 Central coefficients of q in the q-analog of the odd double factorials: a(n) = [q^(n(n-1)/2)] Product_{j=1..n} (1-q^(2j-1))/(1-q).

Original entry on oeis.org

1, 1, 1, 3, 15, 97, 815, 8447, 104099, 1487477, 24188525, 441170745, 8920418105, 198066401671, 4791181863221, 125421804399845, 3532750812110925, 106538929613501939, 3425126166609830467, 116938867144129019137, 4225543021235970185429, 161113285522023566327031

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

			a(n) is the central term of the q-analog of odd double factorials, in which the coefficients of q (triangle A128080) begin:
   n=0: (1);
   n=1: (1);
   n=2: 1,(1),1;
   n=3: 1,2,3,(3),3,2,1;
   n=4: 1,3,6,9,12,14,(15),14,12,9,6,3,1;
   n=5: 1,4,10,19,31,45,60,74,86,94,(97),94,86,74,60,45,31,19,10,4,1;
   n=6: 1,5,15,34,65,110,170,244,330,424,521,614,696,760,801,(815),...;
The terms enclosed in parenthesis are initial terms of this sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A001147 ((2n-1)!!); A128080 (triangle), A128082 (diagonal).

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), q, n*(n-1)/2):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 22 2021

a[n_Integer] := a[n] = Coefficient[Expand@Cancel@FunctionExpand[-q QPochhammer[1/q, q^2, n + 1]/(1 - q)^(n + 1)], q, n (n - 1)/2];
Table[a[n], {n, 0, 21}] (* Vladimir Reshetnikov, Sep 22 2021 *)

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

a(n) ~ 3 * 2^n * n^(n - 3/2) / (sqrt(Pi) * exp(n)). - Vaclav Kotesovec, Feb 07 2023

A128086 A diagonal of the triangle A128084 of coefficients of q in the q-analog of the even double factorials: a(n) = A128084(n,n).

Original entry on oeis.org

1, 1, 2, 7, 24, 86, 315, 1170, 4389, 16588, 63064, 240901, 923858, 3554747, 13716315, 53054703, 205651975, 798645126, 3106669575, 12102626404, 47210910670, 184385864445, 720920510115, 2821499709615, 11052719207369, 43333403693711

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

See A128082 for a diagonal of the triangle A128080 of coefficients of q in the q-analog of the odd double factorials.

			a(n) is the n-th term in the q-analog of even double factorial (2n)!!, in which the coefficients of q (triangle A128084) begin:
(1);
1,(1);
1,2,(2),2,1;
1,3,5,(7),8,8,7,5,3,1;
1,4,9,16,(24),32,39,44,46,44,39,32,24,16,9,4,1;
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.

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

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

A128593 Column 1 of triangle A128592; a(n) = coefficient of q^(n+2) in the q-analog of the odd double factorials (2n+3)!! for n>=0.

Original entry on oeis.org

1, 3, 12, 45, 170, 644, 2451, 9365, 35908, 138104, 532589, 2058782, 7975216, 30951921, 120326060, 468473348, 1826415556, 7129330988, 27860219331, 108984557708, 426730087879, 1672310507262, 6558840830680, 25742937514814, 101108341344396, 397368218111003

Offset: 0

Paul D. Hanna, Mar 12 2007

nonn

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

Cf. A128592; A128080; A001147 ((2n-1)!!); A128594 (column 2), A128595 (row sums).

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+2), q, n+2):
seq(a(n), n=0..30);   # Alois P. Heinz, Sep 22 2021

a[n_] := SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+2}], {q, 0, n+2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 27 2024 *)

{a(n)=polcoeff(prod(j=1,n+2,(1-q^(2*j-1))/(1-q)),n+2,q)}

a(n) = [q^(n+2)] Product_{j=1..n+2} (1-q^(2j-1))/(1-q) for n>=0.

A128592 Triangle, read by rows, of coefficients of q^(nk+k) in the q-analog of the odd double factorials: T(n,k) = [q^(nk+k)] Product_{j=1..n+1} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 45, 97, 45, 1, 1, 170, 696, 696, 170, 1, 1, 644, 4784, 8447, 4784, 644, 1, 1, 2451, 32230, 92003, 92003, 32230, 2451, 1, 1, 9365, 214978, 946330, 1487477, 946330, 214978, 9365, 1, 1, 35908, 1426566, 9417798, 21856230, 21856230

Offset: 0

Paul D. Hanna, Mar 12 2007

			Triangle begins:
1;
1, 1;
1, 3, 1;
1, 12, 12, 1;
1, 45, 97, 45, 1;
1, 170, 696, 696, 170, 1;
1, 644, 4784, 8447, 4784, 644, 1;
1, 2451, 32230, 92003, 92003, 32230, 2451, 1;
1, 9365, 214978, 946330, 1487477, 946330, 214978, 9365, 1;
1, 35908, 1426566, 9417798, 21856230, 21856230, 9417798, 1426566, 35908, 1;
1, 138104, 9441417, 91852376, 302951392, 441170745, 302951392, 91852376, 9441417, 138104, 1;

Cf. A128080; A001147 ((2n-1)!!); A128593 (column 1), A128594 (column 2), A128595 (row sums); variant: A128596.

T[n_, k_] := If[k < 0 || k > n*(n + 1), 0, If[n == 0, 1, SeriesCoefficient[Product[(1 - q^(2*j - 1))/(1 - q), {j, 1, n + 1}], {q, 0, (n + 1)*k}]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 27 2022, from PARI code *)

T(n,k)=if(k<0 || k>n*(n+1),0,if(n==0,1, polcoeff(prod(j=1,n+1,(1-q^(2*j-1))/(1-q)),(n+1)*k,q)))

T(n,k) = A128080(n+1,nk+k) where A128080 is the triangle of coefficients of q in the q-analog of the odd double factorials.

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))}

A128594 Column 2 of triangle A128592; a(n) = coefficient of q^(2n+6) in the q-analog of the odd double factorials (2n+5)!! for n>=0.

Original entry on oeis.org

1, 12, 97, 696, 4784, 32230, 214978, 1426566, 9441417, 62405645, 412278981, 2723566163, 17996243101, 118957645301, 786700165122, 5205396517853, 34461624895701, 228274455988134, 1512920531980961, 10032446308837778

Offset: 0

Paul D. Hanna, Mar 12 2007

nonn

Cf. A128592; A128080; A001147 ((2n-1)!!); A128593 (column 1), A128595 (row sums).

a[n_] := SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+3}], {q, 0, 2n+6}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 27 2024 *)

{a(n)=polcoeff(prod(j=1,n+3,(1-q^(2*j-1))/(1-q)),2*n+6,q)}

a(n) = [q^(2n+6)] Product_{j=1..n+3} (1-q^(2j-1))/(1-q) for n>=0.

A128595 Row sums of triangle A128592.

Original entry on oeis.org

1, 2, 5, 26, 189, 1734, 19305, 253370, 3828825, 65473006, 1249937325, 26352843470, 608142583125, 15247003381854, 412685556908625, 11993673995924378, 372509404162520625, 12313505304343363126, 431620764875678503125

Offset: 0

Paul D. Hanna, Mar 12 2007

nonn

A128592(n,k) is the coefficient of q^(nk+k) in the q-analog of the odd double factorials (2n-1)!!.

Cf. A128592; A128080; A001147 ((2n-1)!!); A128593 (column 1), A128594 (column 2).

a[n_] := Sum[SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+1}], {q, 0, k(n+1)}], {k, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 27 2024 *)

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

a(n) = Sum_{k=0..n} { [q^(nk+k)] Product_{j=1..n+1} (1-q^(2j-1))/(1-q) } for n>=0.

A161123 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k inversions (0 <= k <= n(2n-1)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 6, 0, 9, 0, 12, 0, 14, 0, 15, 0, 14, 0, 12, 0, 9, 0, 6, 0, 3, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 10, 0, 19, 0, 31, 0, 45, 0, 60, 0, 74, 0, 86, 0, 94, 0, 97, 0, 94, 0, 86, 0, 74, 0, 60, 0, 45, 0, 31, 0, 19

Offset: 0

Emeric Deutsch, Jun 05 2009

Sum of entries in row n is (2n-1)!! = A001147(n).
Row n has 1 + 2n(n-1) entries.
Sum_{k>=0} k*T(n,k) = (2n-1)!!*n^2 = A161124(n).
A128080 is the same triangle with the 0's deleted.

			T(3,11)=3 because we have 465132, 546213, and 632541.
Triangle starts:
  1;
  0, 1;
  0, 0, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 2, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1;

Cf. A001147, A128080, A161124.

f := proc (n) options operator, arrow: q^n*(product(1-q^(4*j-2), j = 1 .. n))/(1-q^2)^n end proc: for n from 0 to 4 do P[n] := sort(expand(simplify(f(n)))) end do: for n from 0 to 4 do seq(coeff(P[n], q, j), j = 0 .. n*(2*n-1)) end do; # yields sequence in triangular form

P[n_] := P[n] = q^n*Product[1 - q^(4j - 2), {j, 1, n}]/(1 - q^2)^n // Expand // Simplify;
T[n_, k_] := Coefficient[P[n], q, k];
Table[T[n, k], {n, 0, 5}, {k, 0, n (2n - 1)}] // Flatten (* Jean-François Alcover, Aug 23 2024 *)

Generating polynomial of row n is P_n(q) = (q/(1-q^2))^n*Product_{j=1..n}(1-q^(4j-2)).

cp's OEIS Frontend

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A128084 Triangle, read by rows of n^2+1 terms, of coefficients of q in the q-analog of the even double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A128081 Central coefficients of q in the q-analog of the odd double factorials: a(n) = [q^(n(n-1)/2)] Product_{j=1..n} (1-q^(2j-1))/(1-q).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A128086 A diagonal of the triangle A128084 of coefficients of q in the q-analog of the even double factorials: a(n) = A128084(n,n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A128593 Column 1 of triangle A128592; a(n) = coefficient of q^(n+2) in the q-analog of the odd double factorials (2n+3)!! for n>=0.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A128592 Triangle, read by rows, of coefficients of q^(nk+k) in the q-analog of the odd double factorials: T(n,k) = [q^(nk+k)] Product_{j=1..n+1} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

A128594 Column 2 of triangle A128592; a(n) = coefficient of q^(2n+6) in the q-analog of the odd double factorials (2n+5)!! for n>=0.

Views

Author

Keywords

Crossrefs

Programs