cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Feb 14 2007

Keywords

Comments

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

Examples

			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;
  ...

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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(""))

Formula

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

A181971 Triangle read by rows: T(n,0) = 1, T(n,n) = floor((n+3)/2) and T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 5, 3, 1, 5, 9, 8, 3, 1, 6, 14, 17, 11, 4, 1, 7, 20, 31, 28, 15, 4, 1, 8, 27, 51, 59, 43, 19, 5, 1, 9, 35, 78, 110, 102, 62, 24, 5, 1, 10, 44, 113, 188, 212, 164, 86, 29, 6, 1, 11, 54, 157, 301, 400, 376, 250, 115, 35, 6, 1, 12, 65, 211, 458, 701, 776, 626, 365, 150, 41, 7
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 09 2012

Keywords

Comments

Another variant of Pascal's triangle;
row sums: A081254; central terms: T(2*n,n) = A128082(n+1);
T(n,0) = 1;
T(n,1) = n + 1 for n > 0;
T(n,2) = A000096(n-1) for n > 1;
T(n,3) = A105163(n-2) for n > 2;
T(n,n-2) = A005744(n-1) for n > 1;
T(n,n-1) = A024206(n) for n > 0;
T(n,n) = A008619(n+1).

Examples

			The triangle begins:
.  0:                              1
.  1:                           1     2
.  2:                        1     3     2
.  3:                     1     4     5     3
.  4:                  1     5     9     8     3
.  5:               1     6    14    17    11     4
.  6:            1     7    20    31    28    15     4
.  7:         1     8    27    51    59    43    19     5
.  8:      1     9    35    78   110   102    62    24     5
.  9:   1    10    44   113   188   212   164    86    29     6.

Links

Crossrefs

Cf. A035317, A007318, A074909.

Programs

  • Haskell
    a181971 n k = a181971_tabl !! n !! k
a181971_row n = a181971_tabl !! n
a181971_tabl = map snd $ iterate f (1, [1]) where
   f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i]))
  • Mathematica
    T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = If[n == k, Quotient[n + 3, 2], If[k == 0, 1, If[n > k, T[n - 1, k - 1] + T[n - 1, k]]]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 12 2021 *)
  • PARI
    {T(n,k)=if(n==k,(n+3)\2,if(k==0,1,if(n>k,T(n-1,k-1)+T(n-1,k))))}
for(n=0,12,for(k=0,n,print1(T(n,k),","));print("")) \\ Paul D. Hanna, Jul 18 2012

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

Views

Author

Paul D. Hanna, Feb 14 2007

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Paul D. Hanna, Feb 14 2007

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

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

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

Views

Author

Paul D. Hanna, Feb 14 2007

Keywords

Crossrefs

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

Programs

  • PARI
    {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))}
Showing 1-5 of 5 results.

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