A278339
Number of set partitions of [n] into subsets whose element sums are distinct squares.
Original entry on oeis.org
1, 1, 0, 0, 1, 0, 1, 0, 1, 23, 9, 41, 248, 277, 1556, 2854, 5233, 20701, 145137, 1626890, 4118910, 9963276, 9260756, 64027363, 365237571, 1002679107, 21594036300, 24465529531, 144914973347, 1921444799766
Offset: 0
a(0) = 1: {}.
a(1) = 1: 1.
a(4) = 1: 1|234.
a(6) = 1: 1|2356|4.
a(8) = 1: 12345678.
a(9) = 23: 12345678|9, 123568|4|79, 1236789|45, 1245789|36, 126|345789, 12589|367|4, 1258|3679|4, 12679|358|4, 1267|3589|4, 1345689|27, 135|246789, 13|24568|79, 13579|268|4, 1357|2689|4, 13678|259|4, 13|259|4678, 13|2689|457, 13|268|4579, 156789|234, 18|2345679, 169|23578|4, 1789|2356|4, 178|23569|4.
a(10) = 9: 1|2356|4|78(10)|9, 1|23578|4|6(10)|9, 1|258(10)|367|4|9, 1|258(10)|36|4|79, 1|259|36|4|78(10), 1|267(10)|358|4|9, 1|268|357(10)|4|9, 1|27|3589|4|6(10), 1|27|358|4|69(10).
A299032
Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.
Original entry on oeis.org
1, 1, 0, 3, 6, 0, 12, 106, 420, 2718, 18240, 120879, 694320, 5430438, 40668264, 300401818, 2369504386, 19928714475, 174151735920, 1543284732218, 14224347438876, 135649243229688, 1331658133954940, 13369350846412794, 138122850643702056, 1462610254141337590
Offset: 0
a(4) = 6 because fourth triangular number is 10 and we have [4, 4, 1, 1], [4, 1, 4, 1], [4, 1, 1, 4], [1, 4, 4, 1], [1, 4, 1, 4] and [1, 1, 4, 4].
Cf.
A000217,
A000290,
A066535,
A072964,
A104383,
A126683,
A196010,
A224677,
A224679,
A278340,
A288126,
A298330,
A298858,
A298939,
A299031.
-
b:= proc(n, t) option remember; local i; if n=0 then
`if`(t=0, 1, 0) elif t<1 then 0 else 0;
for i while i^2<=n do %+b(n-i^2, t-1) od; % fi
end:
a:= n-> b(n*(n+1)/2, n):
seq(a(n), n=0..25); # Alois P. Heinz, Feb 05 2018
-
Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 25}]
A299031
Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.
Original entry on oeis.org
1, 1, 0, 3, 18, 60, 252, 1576, 10494, 64152, 458400, 3407019, 27713928, 225193982, 1980444648, 17626414158, 165796077562, 1593587604441, 15985672426992, 163422639872978, 1729188245991060, 18743981599820280, 208963405365941380, 2378065667103672024, 27742569814633730608
Offset: 0
a(3) = 3 because third triangular number is 6 and we have [4, 1, 1], [1, 4, 1] and [1, 1, 4].
Cf.
A000217,
A000290,
A066535,
A072964,
A104383,
A126683,
A196010,
A224677,
A224679,
A278340,
A288126,
A298329,
A298858,
A298938,
A299032.
-
Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]
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}]]
Showing 1-4 of 4 results.