A024940 -id:A024940 - cp's OEIS frontend

A280366 G.f.: Product_{k>=1} (1 + x^(k(k+1)/2)) / (1 - x^(k(k+1)/2)).

1, 2, 2, 4, 6, 6, 10, 14, 14, 20, 28, 30, 38, 50, 54, 66, 86, 94, 110, 138, 152, 178, 218, 238, 274, 330, 362, 412, 488, 534, 602, 710, 778, 864, 1006, 1102, 1220, 1410, 1542, 1696, 1940, 2122, 2328, 2638, 2878, 3148, 3550, 3870, 4214, 4722, 5136, 5580, 6230

Offset: 0

Vaclav Kotesovec, Jan 02 2017

nonn

Convolution of A024940 and A007294.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A024940, A007294, A103265.

nmax = 60; CoefficientList[Series[Product[(1+x^(k*(k+1)/2))/(1-x^(k*(k+1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3 * 2^(-4/3) * Pi^(1/3) * ((2*sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3)) * Zeta(3/2) * (2*sqrt(2)-1) / (32 * sqrt(3) * Pi * n^(3/2)).

A279278 Expansion of Product_{k>=1} (1 + x^(k(k+1)(k+2)/6)).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

Number of partitions of n into distinct tetrahedral numbers (A000292).

			a(35) = 2 because we have [35] and [20, 10, 4, 1].

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

Cf. A000292, A007294, A024940, A068980, A350205 (positions of records).

nmax=120; CoefficientList[Series[Product[1 + x^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).

A279281 Expansion of Product_{k>=1} (1 + x^(k(3k-2))).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

Number of partitions of n into distinct octagonal numbers (A000567).

			a(105) = 2 because we have [96, 8, 1] and [65, 40].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000567, A024940, A033461, A218380, A279041, A279279, A279280.

nmax = 120; CoefficientList[Series[Product[1 + x^(k (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(3*k-2))).

A279279 Expansion of Product_{k>=1} (1 + x^(k(2k-1))).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

Number of partitions of n into distinct hexagonal numbers (A000384).

			a(67) = 2 because we have [66, 1] and [45, 15, 6, 1].

Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000384, A024940, A033461, A218380, A278949, A279280, A279281.

nmax = 120; CoefficientList[Series[Product[1 + x^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(2*k-1))).

A279280 Expansion of Product_{k>=1} (1 + x^(k(5k-3)/2)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 1, 1

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

Number of partitions of n into distinct heptagonal numbers (A000566).

			a(81) = 2 because we have [81] and [55, 18, 7, 1].

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000566, A024940, A033461, A218380, A279012, A279279, A279281.

nmax = 120; CoefficientList[Series[Product[1 + x^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(5*k-3)/2)).

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 0, 0, 1, 1, 1, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 1, 0, 0, 1, 2

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

Number of partitions of n into distinct centered triangular numbers (A005448).

			a(46) = 2 because we have [46] and [31, 10, 4, 1].

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A005448, A024940, A279278, A280950, A281082, A281083, A281084.

nmax = 105; CoefficientList[Series[Product[1 + x^(3 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} (1 + x^(3*k*(k+1)/2+1)).

A307666 Number of partitions of n into consecutive positive triangular numbers.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Ilya Gutkovskiy, Apr 20 2019

nonn

Equivalently, number of ways n can be expressed as the difference between two tetrahedral numbers. - Charlie Neder, Apr 24 2019

Records: a(10)=2, a(2180)=3, a(10053736)=4. - Robert Israel, Aug 20 2019

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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A001227, A007294, A024940, A034706, A296338, A307614, A309783.

N:= 100:
V:= Vector(N):
for i from 1 while i*(i+1)/2 <= N do
  s:= i*(i+1)*(i+2)/6;
  for j from i-1 to 0 by -1 do
    t:= j*(j+1)*(j+2)/6;
    if s-t > N then break fi;
    V[s-t]:= V[s-t]+1
  od;
od:
convert(V,list); # Robert Israel, Aug 20 2019

G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).

A292519 Expansion of Product_{k>=1} 1/(1 + x^(k*(k+1)/2)).

Original entry on oeis.org

1, -1, 1, -2, 2, -2, 2, -2, 2, -2, 1, -1, 2, -1, 1, -3, 3, -3, 4, -4, 5, -6, 5, -6, 8, -6, 6, -8, 6, -6, 7, -5, 6, -7, 5, -7, 9, -7, 9, -11, 9, -11, 13, -10, 12, -15, 12, -14, 16, -13, 15, -15, 11, -14, 15, -11, 15, -18, 15, -19, 23, -21, 25, -27, 24, -28, 28, -24, 28, -29, 24, -28, 31, -25, 29, -33, 30, -35, 36, -35, 42

Offset: 0

Ilya Gutkovskiy, Sep 18 2017

sign

Convolution inverse of A024940.

The difference between the number of partitions of n into an even number of triangular numbers and the number of partitions of n into an odd number of triangular numbers.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for related partition-counting sequences

Cf. A007294, A024940, A280366, A292518.

nmax = 80; CoefficientList[Series[Product[1/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 + x^(k*(k+1)/2)).

A339375 Number of partitions of n into an even number of distinct triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 0, 3, 1, 0, 2, 1, 1, 1, 3, 1, 2, 0, 2, 2, 0, 2, 3, 3, 1, 2, 2, 2, 2, 2, 1, 4, 4, 1, 3, 2, 3, 2, 3, 1, 5, 4, 2, 4, 2, 4, 4, 3, 2, 6, 4, 3, 4, 5, 2, 3, 6, 5, 6, 5, 4, 5, 5, 4, 5, 6, 4

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(31) = 3 because we have [28, 3], [21, 10] and [21, 6, 3, 1].

Cf. A000217, A024940, A067661, A292518, A339366, A339373, A339374, A339376.

nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k (k + 1)/2)), {k, 1, nmax}] + Product[(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k*(k + 1)/2)) + Product_{k>=1} (1 - x^(k*(k + 1)/2))).

a(n) = (A024940(n) + A292518(n)) / 2.

A339376 Number of partitions of n into an odd number of distinct triangular numbers.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 2, 1, 3, 0, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 3, 2, 0, 4, 3, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 3, 4, 4, 4, 1, 5, 4, 2, 3, 5, 3, 6, 4, 2, 6, 4, 3, 5, 6, 5, 5, 5, 5, 5, 4, 5

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(28) = 3 because we have [28], [21, 6, 1] and [15, 10, 3].

Cf. A000217, A024940, A067659, A292518, A339367, A339373, A339374, A339375.

nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k (k + 1)/2)), {k, 1, nmax}] - Product[(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k*(k + 1)/2)) - Product_{k>=1} (1 - x^(k*(k + 1)/2))).

a(n) = (A024940(n) - A292518(n)) / 2.

cp's OEIS Frontend

A280366 G.f.: Product_{k>=1} (1 + x^(k*(k+1)/2)) / (1 - x^(k*(k+1)/2)).

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A279278 Expansion of Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).

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A279281 Expansion of Product_{k>=1} (1 + x^(k*(3*k-2))).

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A279279 Expansion of Product_{k>=1} (1 + x^(k*(2*k-1))).

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A279280 Expansion of Product_{k>=1} (1 + x^(k*(5*k-3)/2)).

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A281081 Expansion of Product_{k>=0} (1 + x^(3*k*(k+1)/2+1)).

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A307666 Number of partitions of n into consecutive positive triangular numbers.

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A292519 Expansion of Product_{k>=1} 1/(1 + x^(k*(k+1)/2)).

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A280366 G.f.: Product_{k>=1} (1 + x^(k(k+1)/2)) / (1 - x^(k(k+1)/2)).

A279278 Expansion of Product_{k>=1} (1 + x^(k(k+1)(k+2)/6)).

A279281 Expansion of Product_{k>=1} (1 + x^(k(3k-2))).

A279279 Expansion of Product_{k>=1} (1 + x^(k(2k-1))).

A279280 Expansion of Product_{k>=1} (1 + x^(k(5k-3)/2)).

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).