A298858 -id:A298858 - cp's OEIS frontend

A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			A000217(4) = 10 = 1 + 3 + 6, so a(4) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000217, A034706, A072964, A196010, A288126, A298858, A307608.

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

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.

A338585 Number of partitions of the n-th triangular number into exactly n positive triangular numbers.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 3, 4, 9, 16, 29, 52, 92, 173, 307, 554, 1002, 1792, 3216, 5738, 10149, 17942, 31769, 55684, 97478, 170356, 295644, 512468, 886358, 1523779, 2614547, 4476152, 7627119, 12966642, 21988285, 37142199, 62591912, 105215149, 176266155, 294591431

Offset: 0

Ilya Gutkovskiy, Nov 08 2020

nonn

			The 5th triangular number is 15 and 15 = 1 + 1 + 1 + 6 + 6 = 3 + 3 + 3 + 3 + 3, so a(5) = 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers
Index entries for sequences related to partitions

Cf. A000217, A007294, A072964, A106337, A196010, A288126, A298858, A307614, A319435, A319797, A319799, A331900, A338586.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i*kn, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))
    end:
a:= n-> (t-> b(t, h(t), n))(n*(n+1)/2):
seq(a(n), n=0..42);  # Alois P. Heinz, Nov 10 2020

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ@Sqrt[8n+1], n, h[n-1]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0], If[i k < n || k > n, 0, b[n, h[i-1], k] + b[n-i, h[Min[n-i, i]], k-1]]];
a[n_] := b[#, h[#], n]&[n(n+1)/2];
a /@ Range[0, 42](* Jean-François Alcover, Nov 15 2020, after Alois P. Heinz *)

# Returns a list of length n, slow.
def GeneralizedEulerTransform(n, a):
    R. = ZZ[[]]
    f = prod((1 - y*x^a(k) + O(x, y)^a(n)) for k in (1..n))
    coeffs = f.inverse().coefficients()
    coeff = lambda k: coeffs[x^a(k)*y^k] if x^a(k)*y^k in coeffs else 0
    return [coeff(k) for k in range(n)]
def A338585List(n): return GeneralizedEulerTransform(n, lambda n: n*(n+1)/2)
print(A338585List(12)) # Peter Luschny, Nov 12 2020

a(n) = [x^A000217(n) y^n] Product_{j>=1} 1 / (1 - y*x^A000217(j)).

a(n) = A319797(A000217(n),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.

A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 3, 13, 3, 55, 201, 159, 865, 1803, 7093, 43431, 14253, 22903, 130851, 120763, 1099693, 4527293, 4976767, 7516897, 14349685, 72866239, 81946383, 167841291, 897853735, 455799253, 946267825, 5054280915, 3941268001, 17066300985, 49111862599

Offset: 0

Ilya Gutkovskiy, Jan 31 2020

nonn

			a(6) = 3 because we have [21], [15, 6] and [6, 15].

Alois P. Heinz, Table of n, a(n) for n = 0..300
Index entries for sequences related to compositions

Cf. A000217, A072964, A126683, A196010, A224677, A288126, A298858, A307614, A331843, A331884.

b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n*(n+1)/2, n, 0):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 31 2020

b[n_, i_, p_] := b[n, i, p] = With[{t = i(i+1)/2}, If[t(i+2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t > n, 0, b[n-t, i-1, p+1]]]]];
a[n_] := b[n(n+1)/2, n, 0];
a /@ Range[0, 37] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

a(n) = A331843(A000217(n)).

More terms from Alois P. Heinz, Jan 31 2020

A335634 Number of ordered ways of writing the n-th n-gonal number as a sum of n nonzero n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 0, 1, 30, 180, 700, 3780, 11844, 50610, 325820, 5803380, 126594910, 2114901789, 28282722650, 323420067880, 3190581939996, 29336527986960, 245438739897312, 1967485926594030, 16000631392009320, 184418174847183508, 4054670001158799616, 111835386569787369559

Offset: 0

Ilya Gutkovskiy, Oct 03 2020

nonn

			a(4) = 1 because the fourth square is 16 and we have [4, 4, 4, 4].

Eric Weisstein's World of Mathematics, Polygonal Number
Index to sequences related to polygonal numbers

Cf. A060354, A298330, A298858, A335633, A337762, A337763, A337764.

Join[{1}, Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 1, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 1, 24}]]

p(n,k) = {k * (k * (n - 2) - n + 4) / 2}
a(n) = {my(m=p(n,n)); polcoef((sum(k=1, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ Andrew Howroyd, Oct 03 2020

a(n) = [x^p(n,n)] (Sum_{k=1..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A338465 Number of ways to write 2*n as an ordered sum of n nonzero triangular numbers.

Original entry on oeis.org

1, 0, 2, 0, 6, 5, 20, 42, 70, 261, 297, 1430, 1584, 7293, 9634, 35945, 60150, 176596, 366401, 886977, 2150421, 4624410, 12205074, 25065216, 67616872, 139894305, 369551925, 793214982, 2011977414, 4517758504, 10992821055, 25669627965, 60531471286, 145112506352

Offset: 0

Ilya Gutkovskiy, Jan 31 2021

nonn

Also number of ways to write n as an ordered sum of n nonnegative numbers one less than a triangular number.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000096, A000217, A106337, A298858, A319799.

Table[SeriesCoefficient[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^n, {x, 0, 2 n}], {n, 0, 33}]

a(n) = [x^(2*n)] (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^n, where theta_2() is the Jacobi theta function.

a(n) = [x^n] (Sum_{k>=0} x^(k*(k + 3)/2))^n.

A336303 Number of ordered ways of writing the n-th n-gonal pyramidal number as a sum of n nonzero n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 0, 0, 6, 0, 180, 630, 1120, 36288, 441000, 6579870, 59734620, 1252872192, 13668490836, 162131872695, 2971275208720, 52783774330940, 1334562954639156, 16933262255752698, 406499325562503480, 8838644883526856832, 190698441426122689290

Offset: 0

Ilya Gutkovskiy, Oct 04 2020

nonn

			a(4) = 6 because the fourth square pyramidal number is 30 and we have [14, 14, 1, 1], [14, 1, 14, 1], [14, 1, 1, 14], [1, 14, 14, 1], [1, 14, 1, 14] and [1, 1, 14, 14].

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A006484, A298858, A335634, A336091, A337797, A337798, A337799.

Join[{1},Table[SeriesCoefficient[Sum[x^(k (k + 1) (k (n - 2) - n + 5)/6), {k, 1, n}]^n, {x, 0, n (n + 1) (n^2 - 3 n + 5)/6}], {n, 1, 22}]]

a(n) = [x^p(n,n)] (Sum_{k=1..n} x^p(n,k))^n, where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.

A338778 Number of ordered ways of writing n-th tetrahedral number as a sum of n positive tetrahedral numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 20, 195, 1890, 6286, 94584, 1065120, 12345432, 194450586, 2844976135, 44569913570, 740023110855, 13144353701940, 241663182769494, 4707408836458200, 95865898167054186, 2038122531703155798, 45103282424247100962, 1037559653596650520776, 24776005985596646165127

Offset: 0

Ilya Gutkovskiy, Nov 08 2020

nonn

			The 5th tetrahedral number is 35 and 35 = 1 + 4 + 10 + 10 + 10 (20 permutations), so a(5) = 20.

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for sequences related to compositions

Cf. A000292, A298269, A298672, A298857, A298858, A303170, A338586.

a(n) = [x^A000292(n)] (Sum_{j>=1} x^A000292(j))^n.

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

Ilya Gutkovskiy, Sep 13 2018

nonn

Index entries for sequences related to sums of squares

Cf. A000217, A000290, A066535, A196010, A232173, A278340, A298858, A299031, A299032.

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}]]

a(n) = [x^(n*(n+1)/2)] theta_3(x)^n, where theta_3() is the Jacobi theta function.

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

cp's OEIS Frontend

A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

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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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A338585 Number of partitions of the n-th triangular number into exactly n positive triangular numbers.

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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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A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

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A335634 Number of ordered ways of writing the n-th n-gonal number as a sum of n nonzero n-gonal numbers.

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A338465 Number of ways to write 2*n as an ordered sum of n nonzero triangular numbers.

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A336303 Number of ordered ways of writing the n-th n-gonal pyramidal number as a sum of n nonzero n-gonal pyramidal numbers.

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