login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k. 2

%I #9 Mar 29 2019 15:51:20

%S 1,1,0,1,2,0,1,4,2,0,1,6,8,4,0,1,8,18,16,6,0,1,10,32,44,32,8,0,1,12,

%T 50,96,102,56,12,0,1,14,72,180,256,216,96,16,0,1,16,98,304,550,624,

%U 428,160,22,0,1,18,128,476,1056,1512,1408,816,256,30,0,1,20,162,704,1862,3240,3820,3008,1494,404,40,0

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k.

%F G.f. of column k: Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k.

%F G.f. of column 2k: (theta_3(x)/theta_4(x))^k, where theta_() is the Jacobi theta function.

%F For asymptotics of column k see comment from _Vaclav Kotesovec_ in A261648.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 2, 4, 6, 8, 10, ...

%e 0, 2, 8, 18, 32, 50, ...

%e 0, 4, 16, 44, 96, 180, ...

%e 0, 6, 32, 102, 256, 550, ...

%e 0, 8, 56, 216, 624, 1512, ...

%t Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i + 1))/(1 - x^(2 i + 1)))^k, {i, 0, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

%t Table[Function[k, SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

%Y Columns k=0-6 give: A000007, A080054, A007096, A261647, A014969, A261648, A014970.

%Y Rows n=0-3 give: A000012, A005843, A001105, A217873.

%Y Main diagonal gives A291697.

%K nonn,tabl

%O 0,5

%A _Ilya Gutkovskiy_, Jul 07 2017

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 09:28 EDT 2024. Contains 371268 sequences. (Running on oeis4.)