A274100 -id:A274100 - cp's OEIS frontend

A274099 Number of partitions of n*(n-1)/2 into at most four parts.

1, 1, 3, 9, 23, 54, 120, 249, 478, 864, 1495, 2484, 3969, 6136, 9234, 13561, 19464, 27378, 37845, 51488, 69012, 91260, 119239, 154078, 197026, 249535, 313290, 390144, 482120, 591519, 720954, 873264, 1051513, 1259130, 1499950, 1778097, 2097984, 2464489

Offset: 1

N. J. A. Sloane, Jun 11 2016

nonn

Colin Barker, Table of n, a(n) for n = 1..1000

A subsequence of A001400. Cf. A274100.

Length[IntegerPartitions[#,4]]&/@Accumulate[Range[0,40]] (* Harvey P. Dale, Jul 08 2022 *)

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
b(n) = round(real((68+36*(-1)^n+18*((-I)^n+I^n)+(16*exp(-2/3*I*n*Pi)*(1+I*sqrt(3)+2*exp((4*I*n*Pi)/3)))/(1+(-1)^(1/3))+59*(1+n)+9*(-1)^n*(1+n)+18*(1+n)*(2+n)+2*(1+n)*(2+n)*(3+n))/288))
vector(50, n, b(n*(n-1)/2)) \\ Colin Barker, Jun 12 2016

Coefficient of x^(n*(n-1)/2) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

Empirical g.f.: (1 -5*x +15*x^2 -30*x^3 +54*x^4 -77*x^5 +109*x^6 -128*x^7 +150*x^8 -148*x^9 +150*x^10 -128*x^11 +109*x^12 -77*x^13 +54*x^14 -30*x^15 +15*x^16 -5*x^17 +x^18) / ((1 -x)^7*(1 +x^2)^3*(1 +x +x^2)*(1 +x^4)). - Colin Barker, Jun 12 2016

More terms from Colin Barker, Jun 12 2016

A274232 Number of partitions of 2^n into at most three parts.

Original entry on oeis.org

1, 2, 4, 10, 30, 102, 374, 1430, 5590, 22102, 87894, 350550, 1400150, 5596502, 22377814, 89494870, 357946710, 1431721302, 5726754134, 22906754390, 91626493270, 366504924502, 1466017600854, 5864066209110, 23456256447830, 93825009014102, 375300002501974

Offset: 0

Colin Barker, Jun 15 2016

nonn

Colin Barker, Table of n, a(n) for n = 0..1000

A subsequence of A001399. Cf. A274100, A274233.

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)).
b(n) = round(real((47+9*(-1)^n + 8*exp(-2/3*I*n*Pi) + 8*exp((2*I*n*Pi)/3) + 36*n+6*n^2)/72))
vector(50, n, n--; b(2^n))

Coefficient of x^(2^n) in 1/((1-x)*(1-x^2)*(1-x^3)).
Conjectures: (Start)
a(n) = (8+3*2^(1+n)+4^n)/12 for n>0.
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-5*x+4*x^2+2*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)

A274271 Number of partitions of 3^n into at most four parts.

Original entry on oeis.org

1, 3, 18, 225, 4410, 105903, 2746098, 73140525, 1965803130, 52995903003, 1430162760978, 38607856205625, 1042353276205050, 28143008896575303, 759856474192364658, 20516081909157771525, 553933825501236490170, 14956209814120079146803, 403817633711525094117138

Offset: 0

Colin Barker, Jun 17 2016

nonn

Colin Barker, Table of n, a(n) for n = 0..650

A subsequence of A001400.

Cf. A274100 (2^n), A274272 (5^n).

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
b(n) = round(real(68+36*(-1)^n+18*((-I)^n+I^n)+(16*exp(-2/3*I*n*Pi)*(1+I*sqrt(3)+2*exp((4*I*n*Pi)/3)))/(1+(-1)^(1/3))+59*(1+n)+9*(-1)^n*(1+n)+18*(1+n)*(2+n)+2*(1+n)*(2+n)*(3+n))/288)
vector(20, n, n--; b(3^n))

Coefficient of x^(3^n) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
Conjectures (Start)
a(n) = ((3+3^n)^2*(9+3^n))/144 for n>1.
a(n) = 40*a(n-1)-390*a(n-2)+1080*a(n-3)-729*a(n-4) for n>4.
G.f.: (1-37*x+288*x^2-405*x^3-81*x^4) / ((1-x)*(1-3*x)*(1-9*x)*(1-27*x)).
(End)

A274272 Number of partitions of 5^n into at most four parts.

Original entry on oeis.org

1, 6, 185, 15246, 1736385, 212946246, 26516391385, 3312004971246, 413937039016385, 51740540399346246, 6467527813385891385, 808439983261977471246, 101054973072475964016385, 12631871013177766274346246, 1578983861125177809948391385

Offset: 0

Colin Barker, Jun 17 2016

nonn

Colin Barker, Table of n, a(n) for n = 0..450

A subsequence of A001400.

Cf. A274100 (2^n), A274271 (3^n).

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
b(n) = round(real(68+36*(-1)^n+18*((-I)^n+I^n)+(16*exp(-2/3*I*n*Pi)*(1+I*sqrt(3)+2*exp((4*I*n*Pi)/3)))/(1+(-1)^(1/3))+59*(1+n)+9*(-1)^n*(1+n)+18*(1+n)*(2+n)+2*(1+n)*(2+n)*(3+n))/288)
vector(20, n, n--; b(5^n))

Coefficient of x^(5^n) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
Conjectures (Start)
a(n) = (57+8*(-1)^n+63*5^n+3*5^(1+2*n)+125^n)/144.
a(n) = 155*a(n-1)-3874*a(n-2)+15470*a(n-3)+3875*a(n-4)-15625*a(n-5) for n>4.
G.f.: (1-149*x+3129*x^2-5655*x^3-6750*x^4) / ((1-x)*(1+x)*(1-5*x)*(1-25*x)*(1-125*x)).
(End)

cp's OEIS Frontend

A274099 Number of partitions of n*(n-1)/2 into at most four parts.

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A274232 Number of partitions of 2^n into at most three parts.

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A274271 Number of partitions of 3^n into at most four parts.

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A274272 Number of partitions of 5^n into at most four parts.

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