A224677
Number of compositions of n*(n+1)/2 into sums of positive triangular numbers.
Original entry on oeis.org
1, 1, 2, 7, 40, 351, 4876, 104748, 3487153, 179921982, 14387581923, 1783124902639, 342504341570010, 101962565961894431, 47044167891731682278, 33640402686770010577421, 37282664267078280296013183, 64038780633654058635677191329, 170478465430659361252118580217675
Offset: 0
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b:= proc(n) option remember; local i; if n=0 then 1 else 0;
for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; % fi
end:
a:= n-> b(n*(n+1)/2):
seq(a(n), n=0..20); # Alois P. Heinz, Feb 05 2018
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b[n_] := b[n] = If[n==0, 1, Sum[If[IntegerQ[Sqrt[8j+1]], b[n-j], 0], {j, 1, n}]];
a[n_] := b[n(n+1)/2];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz in A023361 *)
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{a(n)=polcoeff(1/(1-sum(r=1,n+1, x^(r*(r+1)/2)+x*O(x^(n*(n+1)/2)))), n*(n+1)/2)}
for(n=0, 20, print1(a(n), ", "))
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.
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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.
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Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]
Showing 1-3 of 3 results.