A104383 -id:A104383 - cp's OEIS frontend

A066655 Number of partitions of n*(n-1)/2.

1, 1, 3, 11, 42, 176, 792, 3718, 17977, 89134, 451276, 2323520, 12132164, 64112359, 342325709, 1844349560, 10015581680, 54770336324, 301384802048, 1667727404093, 9275102575355, 51820051838712, 290726957916112, 1637293969337171, 9253082936723602

Offset: 1

Roberto E. Martinez II, Jan 10 2002

nonn

Number of partitions of the number of edges of the complete graph of order n, K_n.

			a(4) = p(6) = 11.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000041, A000217, A007294, A104383.

Cf. A173519. - Reinhard Zumkeller, Feb 20 2010

Table[PartitionsP[n(n-1)/2], {n, 1, 30}]

combinat::partitions::count(binomial(n+2,n)) $n=-1..40 // Zerinvary Lajos, Apr 16 2007

a(n) = numbpart(n*(n-1)/2); \\ Michel Marcus, Dec 18 2017

a(n) = p(n*(n-1)/2) = A000041(n*(n-1)/2).
a(n) ~ exp(Pi*sqrt(n*(n-1)/3))/(2*sqrt(3)*n*(n - 1)). - Ilya Gutkovskiy, Jan 13 2017
a(n) ~ exp(Pi*(n - 1/2) / sqrt(3)) / (2*sqrt(3)*n^2). - Vaclav Kotesovec, May 17 2018

More terms from Vladeta Jovovic, Jan 12 2002

Edited by Dean Hickerson, Jan 14 2002

A104382 Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 12, 6, 1, 1, 10, 27, 27, 10, 1, 1, 13, 52, 84, 57, 14, 1, 1, 17, 91, 206, 221, 110, 21, 1, 1, 22, 147, 441, 674, 532, 201, 29, 1, 1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1, 1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1, 1, 38, 469

Offset: 1

Paul D. Hanna, Mar 04 2005

Secondary diagonal equals partitions of n - 1 (A000065).
Third diagonal is A104384.
Third column is A104385.
Row sums are A104383 where limit_{n --> inf} A104383(n+1)/A104383(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...

			Rows begin:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 7, 12, 6, 1;
1, 10, 27, 27, 10, 1;
1, 13, 52, 84, 57, 14, 1;
1, 17, 91, 206, 221, 110, 21, 1;
1, 22, 147, 441, 674, 532, 201, 29, 1;
1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1;
1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1; ...

Abramowitz, M. and Stegun, I. A. (Editors). "Partitions into Distinct Parts." S24.2.2 in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, pp. 825-826, 1972.

Alois P. Heinz, Rows n = 1..55, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Partition Function Q.

Cf. A008289, A000009, A000065, A104383, A104384, A104385.

PARI
```
T(n,k)=if(n
				
```

T(n, 1) = T(n, n) = 1.
T(n, n-1) = A000065(n).
T(n, 2) = [(n*(n+1)/2-1)/2].
From Álvar Ibeas, Jul 23 2020: (Start)
T(n, k) = A008284((n-k+1)*(n+k)/2, k).
T(n, k) = A026820((n-k)*(n+k+1)/2, k), with A026820(0, k) = 1. (End)

A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 33, 68, 144, 312, 686, 1523, 3405, 7652, 17284, 39246, 89552, 205253, 472297, 1090544, 2525904, 5867037, 13663248, 31896309, 74628130, 174972341, 411032475, 967307190, 2280248312, 5383723722, 12729879673, 30141755384, 71462883813

Offset: 0

Moshe Shmuel Newman, Feb 15 2007

nonn

Also the number of self-conjugate partitions of the n-th triangular number.

			The 5th triangular number is 15. Writing this as a sum of distinct odd numbers: 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3 are all the possibilities. So a(5) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Sequences A066655 and A104383 do the same thing for triangular numbers, with partitions or distinct partitions. Sequences A072213 and A072243 are analogs for squares rather than triangular numbers.

Cf. A000217.

g:= mul(1+x^(2*j+1),j=0..900): seq(coeff(g,x,n*(n+1)/2),n=0..40); # Emeric Deutsch, Feb 27 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i^2n, 0, b(n-2*i+1, i-1))))
    end:
a:= n-> b(n*(n+1)/2, ceil(n*(n+1)/4)*2-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 31 2018

a[n_] := SeriesCoefficient[QPochhammer[-x, x^2], {x, 0, n*(n+1)/2}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 25 2018 *)

More terms from Emeric Deutsch, Feb 27 2007

a(0)=1 prepended by Alois P. Heinz, Jan 31 2018

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

Ilya Gutkovskiy, Feb 01 2018

nonn

			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}]

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k^2))^n.

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

Ilya Gutkovskiy, Feb 01 2018

nonn

			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}]

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=0} x^(k^2))^n.

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A066655 Number of partitions of n*(n-1)/2.

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A104382 Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.

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A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

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A299032 Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.

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A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.

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