A308422 -id:A308422 - cp's OEIS frontend

A226141 Sum of the squared parts of the partitions of n into exactly two parts.

0, 2, 5, 18, 30, 64, 91, 156, 204, 310, 385, 542, 650, 868, 1015, 1304, 1496, 1866, 2109, 2570, 2870, 3432, 3795, 4468, 4900, 5694, 6201, 7126, 7714, 8780, 9455, 10672, 11440, 12818, 13685, 15234, 16206, 17936, 19019, 20940, 22140, 24262, 25585, 27918, 29370, 31924, 33511

Offset: 1

Wesley Ivan Hurt, May 27 2013

			a(5) = 30; 5 has exactly 2 partitions into two parts, (4,1) and (3,2). Squaring the parts and adding, we get: 1^2 + 2^2 + 3^2 + 4^2 = 30.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), this sequence (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Cf. A000330, A308422.

[n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8 : n in [1..80]]; // Wesley Ivan Hurt, Jun 22 2024

a:=n->sum(i^2 + (n-i)^2, i=1..floor(n/2)); seq((a(k), k=1..40);

Array[Sum[i^2 + (# - i)^2, {i, Floor[#/2]}] &, 39] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,2,5,18,30,64,91},50] (* Harvey P. Dale, Jul 23 2019 *)

a(n) = Sum_{i=1..floor(n/2)} (i^2 + (n-i)^2).
a(n) = n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8. - Giovanni Resta, May 29 2013
G.f.: x^2*(2+3*x+7*x^2+3*x^3+x^4) / ( (1+x)^3*(x-1)^4 ). - R. J. Mathar, Jun 07 2013
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7). - Wesley Ivan Hurt, Jun 22 2024
a(n) = A000330(n) - A308422(n). - Wesley Ivan Hurt, Jul 16 2025

A309335 a(n) = n^3 if n odd, 7*n^3/8 if n even.

Original entry on oeis.org

0, 1, 7, 27, 56, 125, 189, 343, 448, 729, 875, 1331, 1512, 2197, 2401, 3375, 3584, 4913, 5103, 6859, 7000, 9261, 9317, 12167, 12096, 15625, 15379, 19683, 19208, 24389, 23625, 29791, 28672, 35937, 34391, 42875, 40824, 50653, 48013, 59319, 56000, 68921, 64827, 79507, 74536, 91125

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A007331.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A000578, A007331, A016755, A026741, A059376, A248882, A303383 (partial sums), A308422, A309336.

a[n_] := If[OddQ[n], n^3, 7 n^3/8]; Table[a[n], {n, 0, 45}]
nmax = 45; CoefficientList[Series[x (1 + 7 x + 23 x^2 + 28 x^3 + 23 x^4 + 7 x^5 + x^6)/(1 - x^2)^4, {x, 0, nmax}], x]
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 7, 27, 56, 125, 189, 343}, 46]
Table[n^3 (15 - (-1)^n)/16, {n, 0, 45}]

G.f.: x * (1 + 7*x + 23*x^2 + 28*x^3 + 23*x^4 + 7*x^5 + x^6)/(1 - x^2)^4.
G.f.: Sum_{k>=1} J_3(k) * x^k/(1 - x^(2*k)), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) * (1 - 1/2^s).
a(n) = n^3 * (15 - (-1)^n)/16.
a(n) = Sum_{d|n, n/d odd} J_3(d).
Sum_{n>=1} 1/a(n) = 57*zeta(3)/56 = 1.223522205001729897639...
Multiplicative with a(2^e) = 7*2^(3*e-3), and a(p^e) = p^(3*e) for odd primes p. - Amiram Eldar, Oct 26 2020
Euler transform is A248882. - Georg Fischer, Nov 10 2020

A308418 Expansion of e.g.f. exp(x(1 + 3x + 6x^2 + 3x^3 + x^4)/(1 - x^2)^3).

Original entry on oeis.org

1, 1, 7, 73, 649, 8821, 122311, 2064637, 37933393, 773276329, 17257075111, 414876953041, 10780187135257, 298418920103773, 8812636845668839, 275368711393020421, 9091457478119636641, 315782978460465185617, 11511089733834178827463, 439231563093877354663129

Offset: 0

Ilya Gutkovskiy, May 25 2019

nonn

Cf. A007434, A059376, A088009, A255807, A308417, A308422.

nmax = 19; CoefficientList[Series[Exp[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Product[(1 + x^k)^(DirichletConvolve[j^3, MoebiusMu[j], j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[1/8 (7 - (-1)^k) k^2 k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

my(x ='x + O('x^30)); Vec(serlaplace(exp(x*(1+3*x+6*x^2+3*x^3+x^4)/(1-x^2)^3))) \\ Michel Marcus, May 26 2019

E.g.f.: exp(Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k))), where J_2() is the Jordan function (A007434).
E.g.f.: Product_{k>=1} (1 + x^k)^(J_3(k)/k), where J_3() is the Jordan function (A059376).
a(n) ~ 2^(-5/4) * 21^(1/8) * n^(n - 1/8) * exp(2^(3/2) * 3^(-3/4) * 7^(1/4) * n^(3/4) - n). - Vaclav Kotesovec, May 28 2019
E.g.f.: exp(Sum_{k>=1} A308422(k)*x^k). - Ilya Gutkovskiy, May 29 2019

A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

Original entry on oeis.org

0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561, 9375, 14641, 19440, 28561, 36015, 50625, 61440, 83521, 98415, 130321, 150000, 194481, 219615, 279841, 311040, 390625, 428415, 531441, 576240, 707281, 759375, 923521, 983040, 1185921, 1252815, 1500625, 1574640, 1874161, 1954815

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A285989.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000583, A016756, A026741, A059377, A285989, A308422, A309335.

a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 15 x + 76 x^2 + 165 x^3 + 230 x^4 + 165 x^5 + 76 x^6 + 15 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]
Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]

G.f.: x * (1 + 15*x + 76*x^2 + 165*x^3 + 230*x^4 + 165*x^5 + 76*x^6 + 15*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 - x^(2*k)), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 1/2^s).
a(n) = n^4 * (31 - (-1)^n)/32.
a(n) = Sum_{d|n, n/d odd} J_4(d).
Sum_{n>=1} 1/a(n) = 241*Pi^4/21600 = 1.086832913851601267313987...
Multiplicative with a(2^e) = 15*2^(4*e-4), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020

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A226141 Sum of the squared parts of the partitions of n into exactly two parts.

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A309335 a(n) = n^3 if n odd, 7*n^3/8 if n even.

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A308418 Expansion of e.g.f. exp(x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3).

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A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

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A308418 Expansion of e.g.f. exp(x(1 + 3x + 6x^2 + 3x^3 + x^4)/(1 - x^2)^3).