The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function. 22
 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 29, 17, 6, 1, 1, 7, 36, 70, 54, 26, 7, 1, 1, 8, 63, 157, 165, 99, 35, 8, 1, 1, 9, 106, 337, 482, 357, 163, 45, 9, 1, 1, 10, 171, 702, 1319, 1203, 688, 239, 58, 10, 1, 1, 11, 265, 1420, 3390, 3819, 2673, 1154, 344, 73, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS A(n,k) is the number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_k)^2 <= n^2. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1274 Eric Weisstein's World of Mathematics, Jacobi Theta Functions FORMULA A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j>=0} x^(j^2))^k. EXAMPLE Square array begins:   1,  1,   1,   1,    1,     1,  ...   1,  2,   3,   4,    5,     6,  ...   1,  3,   6,  11,   20,    36,  ...   1,  4,  11,  29,   70,   157,  ...   1,  5,  17,  54,  165,   482,  ...   1,  6,  26,  99,  357,  1203,  ... MATHEMATICA Table[Function[k, SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^k/(2^k (1 - x)), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, 0, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten PROG (PARI) T(n, k)={if(k==0, 1, polcoef(((sum(j=0, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019 CROSSREFS Columns k=0..10 give A000012, A000027, A000603, A000604, A055403, A055404, A055405, A055406, A055407, A055408, A055409. Rows n=0..10 give A000012, A000027, A055417, A055418, A055419, A055420, A055421, A055422, A055423, A055424, A055425. Main diagonal gives A302863. Cf. A000122, A122510, A302996, A302997. Sequence in context: A230861 A119724 A162424 * A303484 A008571 A230860 Adjacent sequences:  A302995 A302996 A302997 * A302999 A303000 A303001 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, Apr 17 2018 STATUS approved

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

Last modified April 17 05:12 EDT 2021. Contains 343059 sequences. (Running on oeis4.)