A196010 -id:A196010 - cp's OEIS frontend

A298858 Number of ordered ways of writing n-th triangular number as a sum of n nonzero triangular numbers.

1, 1, 0, 0, 4, 11, 86, 777, 4670, 36075, 279482, 2345201, 21247326, 197065752, 1983741228, 20769081251, 228078253168, 2604226354265, 30880251148086, 379415992755572, 4818158748326064, 63116999199457944, 851467484377802094, 11811530978240316682, 168243449082524484856

Offset: 0

Ilya Gutkovskiy, Jan 27 2018

nonn

			a(4) = 4 because fourth triangular number is 10 and we have [3, 3, 3, 1], [3, 3, 1, 3], [3, 1, 3, 3] and [1, 3, 3, 3].

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

Cf. A000217, A007294, A023361, A024940, A072964, A106337, A196010, A288126, A298329, A298330.

Table[SeriesCoefficient[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]

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

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

Original entry on oeis.org

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).

A232108 a(n) = [x^(n(n+1)/2)] G(x)^(n+1) where G(x) = Sum_{n>=0} x^(n(n+1)/2).

Original entry on oeis.org

1, 2, 4, 14, 90, 438, 3151, 24390, 204156, 1833212, 17301306, 175936764, 1870247133, 20872753540, 243478609605, 2957875659062, 37319273049382, 487266892836348, 6574891059415183, 91475580555526776, 1309960647920094337, 19278546942842385994, 291167370195970990704, 4507447478297070537800

Offset: 0

Paul D. Hanna, Nov 18 2013

nonn

			Let G(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
then a(n) = the coefficient of x^(n*(n+1)/2) in G(x)^n.
Coefficients of x^k in powers of G(x)^n begin:
n\k...0...1..2..3..4..5...6...7...8...9..10..11..12...13..14...15...16...
n=1: [(1),1, 0, 1, 0, 0,  1,  0,  0,  0,  1,  0,  0,   0,  0,   1,   0,...];
n=2: [1, (2),1, 2, 2, 0,  3,  2,  0,  2,  2,  2,  1,   2,  0,   2,   4,...];
n=3: [1,  3, 3,(4),6, 3,  6,  9,  3,  7,  9,  6,  9,   9,  6,   6,  15,...];
n=4: [1,  4, 6, 8,13,12,(14),24, 18, 20, 32, 24, 31,  40, 30,  32,  48,...];
n=5: [1,  5,10,15,25,31, 35, 55, 60, 60,(90),90, 95, 135,125, 126, 170,...];
n=6: [1,  6,15,26,45,66, 82,120,156,170,231,276,290, 390,435,(438),561,...]; ...
the coefficients in parenthesis form the initial terms of this sequence.

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A196010.

{a(n)=local(G=sum(m=0, n+1, x^(m*(m+1)/2))+x*O(x^(n*(n+1)/2))); polcoeff(G^(n+1), n*(n+1)/2)}
for(n=0,30,print1(a(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

A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).

Original entry on oeis.org

1, 1, 3, 6, 5, 95, 336, 2597, 26832, 197577, 1847800, 14621101, 129754956, 1146534701, 12342194879, 161225146370, 2464561564936, 39642413790129, 620059254486798, 9430493858327959, 136438759335452360, 1881721996407396801, 24999081626667425376, 321601467988647184779

Offset: 0

Ilya Gutkovskiy, Oct 03 2020

nonn

			a(3) = 6 because the third triangular number is 6 and we have [6, 0, 0], [0, 6, 0], [0, 0, 6], [3, 3, 0], [3, 0, 3] and [0, 3, 3].

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

Cf. A060354, A088218, A196010, A298329, A335634, A337762, A337763, A337764.

Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 0, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 0, 23}]

p(n,k) = {k * (k * (n - 2) - n + 4) / 2}
a(n) = {my(m=p(n,n)); polcoef((sum(k=0, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ Andrew Howroyd, Oct 03 2020

a(n) = [x^p(n,n)] (Sum_{k=0..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A336091 Number of ordered ways of writing the n-th n-gonal pyramidal number as a sum of n n-gonal pyramidal numbers (with 0's allowed).

Original entry on oeis.org

1, 1, 2, 3, 10, 5, 246, 1519, 19678, 74601, 690490, 21026621, 301528272, 4397123315, 71221546592, 1001245733295, 19276579678736, 368677642975493, 6820451221691646, 136000924000323691, 3069656935024721420, 69646109074231173897, 1641880679174919030100

Offset: 0

Ilya Gutkovskiy, Oct 04 2020

nonn

			a(3) = 3 because the third tetrahedral (or triangular pyramidal) number is 10 and we have [10, 0, 0], [0, 10, 0] and [0, 0, 10].

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

Cf. A006484, A196010, A335633, A336303, A337797, A337798, A337799.

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

a(n) = [x^p(n,n)] (Sum_{k=0..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.

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

A298858 Number of ordered ways of writing n-th triangular number as a sum of n nonzero triangular numbers.

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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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A232108 a(n) = [x^(n*(n+1)/2)] G(x)^(n+1) where G(x) = Sum_{n>=0} x^(n*(n+1)/2).

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PARI

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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Maple

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A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).

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A232108 a(n) = [x^(n(n+1)/2)] G(x)^(n+1) where G(x) = Sum_{n>=0} x^(n(n+1)/2).