A319935
T(n,k) = [x^n] JacobiTheta3(0,x)^k, for 0 <= k <= n, triangle read by rows.
Original entry on oeis.org
1, 0, 2, 0, 0, 4, 0, 0, 0, 8, 0, 2, 4, 6, 24, 0, 0, 8, 24, 48, 112, 0, 0, 0, 24, 96, 240, 544, 0, 0, 0, 0, 64, 320, 960, 2368, 0, 0, 4, 12, 24, 200, 1020, 3444, 9328, 0, 2, 4, 30, 104, 250, 876, 3542, 12112, 34802, 0, 8, 24, 144, 560, 1560, 4424, 14112, 44640, 129064, 339064
Offset: 0
Triangle starts:
[0] 1
[1] 0, 2
[2] 0, 0, 4
[3] 0, 0, 0, 8
[4] 0, 2, 4, 6, 24
[5] 0, 0, 8, 24, 48, 112
[6] 0, 0, 0, 24, 96, 240, 544
[7] 0, 0, 0, 0, 64, 320, 960, 2368
[8] 0, 0, 4, 12, 24, 200, 1020, 3444, 9328
[9] 0, 2, 4, 30, 104, 250, 876, 3542, 12112, 34802
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A319935row := proc(n) local ser;
ser := j -> series(JacobiTheta3(0, x)^j, x, n+1);
seq(coeff(ser(j), x, n), j=0..n) end:
seq(A319935row(n), n=0..10);
A294071
Number of ordered ways of writing n^2 as a sum of n squares > 1.
Original entry on oeis.org
1, 0, 0, 0, 1, 5, 6, 7, 288, 262, 13702, 69531, 610567, 5356091, 51724960, 521956086, 5467658641, 59931636545, 690518644584, 8100858045744, 99142980567486, 1246972499954475, 16142015005905558, 215722810653380845, 2955759897694815985, 41614888439136252691
Offset: 0
a(5) = 5 because we have [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].
Cf.
A000290,
A037444,
A066535,
A078134,
A092362,
A232173,
A280129,
A280542,
A281154,
A281155,
A298329,
A298330,
A298640,
A298642.
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Table[SeriesCoefficient[((-1 - 2 x + EllipticTheta[3, 0, x])/2)^n, {x, 0, n^2}], {n, 0, 25}]
A319221
Number of ordered ways of writing n-th triangular number as a sum of n squares.
Original entry on oeis.org
1, 2, 0, 24, 144, 960, 4608, 74048, 859952, 9568800, 109975680, 1647979872, 23917274304, 358378620704, 5528847787008, 94307761212304, 1632598198916544, 29205907283227776, 538335591996965760, 10388234139989630128, 205386383159397554688, 4173254005731822569088
Offset: 0
-
Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n, {x, 0, n (n + 1)/2}], {n, 0, 21}]
Join[{1}, Table[SquaresR[n, n (n + 1)/2], {n, 21}]]
A319223
Number of ordered ways of writing n^3 as a sum of n squares.
Original entry on oeis.org
1, 2, 4, 32, 24, 14112, 674368, 39801344, 2454266992, 166591027058, 12820702401872, 1156778646258336, 119773060481140800, 14004241350957965408, 1791476464655904407168, 247572699435320047056384, 36696694077934168215974368, 5825316759916541565549586176, 989291135292653632945527984868
Offset: 0
-
Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n, {x, 0, n^3}], {n, 0, 18}]
Join[{1}, Table[SquaresR[n, n^3], {n, 18}]]