A000294 -id:A000294 - cp's OEIS frontend

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

1, 1, 3, 10, 40, 150, 616, 2456, 10102, 41400, 171526, 712111, 2972115, 12434993, 52195414, 219567909, 925704792, 3909841659, 16541598215, 70085877919, 297347922785, 1263046810334, 5370930049915, 22861883482838, 97402827429118, 415332438952517, 1772380322197432

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A000294, A023871, A274998, A278767, A278768, A278769.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = 4.3505530790182509701639869563721679988879373943131559534408716195123... and c = 0.2276354216252041005336767937139336687746108521151301186102034... - Vaclav Kotesovec, Aug 18 2018

A264923 G.f.: 1 / Product_{n>=0} (1 - x^(n+3))^((n+1)*(n+2)/2!).

Original entry on oeis.org

1, 0, 0, 1, 3, 6, 11, 18, 33, 57, 105, 183, 330, 567, 990, 1693, 2904, 4917, 8343, 14010, 23511, 39171, 65100, 107592, 177352, 290931, 475905, 775381, 1259637, 2039094, 3291613, 5296467, 8499339, 13599292, 21702795, 34541724, 54839894, 86847255, 137212197, 216274466, 340129773, 533726442, 835732774, 1305877914, 2036369010

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 3 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 18*x^7 + 33*x^8 + 57*x^9 + 105*x^10 +...
where
1/A(x) = (1-x^3) * (1-x^4)^3 * (1-x^5)^6 * (1-x^6)^10 * (1-x^7)^15 * (1-x^8)^21 * (1-x^9)^28 * (1-x^10)^36 * (1-x^11)^45 *...
Also,
log(A(x)) = (x/(1-x))^3 + (x^2/(1-x^2))^3/2 + (x^3/(1-x^3))^3/3 + (x^4/(1-x^4))^3/4 + (x^5/(1-x^5))^3/5 + (x^6/(1-x^6))^3/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000

Cf. A052847, A264924, A264925, A264926.

Cf. A000294, A217093, A258349.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-2)*(k-1)/2), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+3) +x*O(x^n) )^((k+1)*(k+2)/2) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^3 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)/2! )}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^3 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)/2!.
a(n) ~ Pi^(3/8) / (2^(55/32) * 15^(7/32) * n^(23/32)) * exp(29*Zeta(3)/(8*Pi^2) - log(2*Pi)/2 - 3*Zeta'(-1)/2 - 2025*Zeta(3)^3/(2*Pi^8) + (5^(1/4)*Pi/6^(3/4) - 135*15^(1/4)*Zeta(3)^2/(2^(7/4)*Pi^5)) * n^(1/4) - 3*sqrt(15*n/2)*Zeta(3)/Pi^2 + 2^(7/4)*Pi/(3*15^(1/4)) * n^(3/4)). - Vaclav Kotesovec, Dec 09 2015

A278403 a(n) = Sum_{d|n} d^2 * (d+1)/2.

Original entry on oeis.org

1, 7, 19, 47, 76, 151, 197, 335, 424, 632, 727, 1127, 1184, 1673, 1894, 2511, 2602, 3634, 3611, 4872, 5066, 6299, 6349, 8615, 8201, 10316, 10630, 13081, 12616, 16526, 15377, 19407, 19258, 22838, 22322, 28586, 26012, 31775, 31622, 37960, 35302, 44594, 40679, 49899, 48874, 56081, 53017, 67239, 60222, 72507, 70246, 82012, 75844, 94030, 85502, 102745, 97850, 111860, 104431, 131502

Offset: 1

Paul D. Hanna, Nov 20 2016

			L.g.f.: L(x) = x + 7*x^2/2 + 19*x^3/3 + 47*x^4/4 + 76*x^5/5 + 151*x^6/6 + 197*x^7/7 + 335*x^8/8 + 424*x^9/9 + 632*x^10/10 + 727*x^11/11 + 1127*x^12/12 +...
which equals the series
L(x) = x/(1-x)^3 + (x^2/2)/(1-x^2)^3 + (x^3/3)/(1-x^3)^3 + (x^4/4)/(1-x^4)^3 + (x^5/5)/(1-x^5)^3 + (x^6/6)/(1-x^6)^3 + (x^7/7)/(1-x^7)^3 +...
The exponentiation of the l.g.f. equals the infinite product
exp(L(x)) = 1/((1-x)*(1-x^2)^3*(1-x^3)^6*(1-x^4)^10*(1-x^5)^15*(1-x^6)^21*...);
explicitly,
exp(L(x)) = 1 + x + 4*x^2 + 10*x^3 + 26*x^4 + 59*x^5 + 141*x^6 + 310*x^7 + 692*x^8 + 1483*x^9 + 3162*x^10 + 6583*x^11 + 13602*x^12 +...+ A000294(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A000294, A001157, A001158, A277613.

Table[Total[#^2*(#+1)/2&/@Divisors[n]],{n,60}] (* Harvey P. Dale, Jul 26 2017 *)

{a(n) = sumdiv(n,d,d^2*(d+1)/2)}
for(n=1,60,print1(a(n),", "))

{a(n) = (sigma(n,3) + sigma(n,2))/2}
for(n=1,60,print1(a(n),", "))

{a(n) = n * polcoeff( sum(k=1,n, (x^k/k) / (1 - x^k +x*O(x^n))^3), n)}
for(n=1,60,print1(a(n),", "))

Let the l.g.f. be L(x) = Sum_{n>=1} a(n)*x^n/n, then:
(1) exp( L(x) ) = Product_{n>=1} 1/(1 - x^n)^(n*(n+1)/2),
(2) L(x) =  Sum_{n>=1} (x^n/n) / (1 - x^n)^3.
O.g.f.: Sum_{n>=1} n^2*(n+1)/2 * x^n / (1 - x^n).
a(n) = (sigma_3(n) + sigma_2(n))/2, where sigma_2(n) = A001157(n) and sigma_3(n) = A001158(n).
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 720. - Vaclav Kotesovec, Jul 13 2021
Dirichlet g.f.: zeta(s) * (zeta(s-3) + zeta(s-2)) / 2. - Amiram Eldar, Jan 02 2025

A292386 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)/2).

Original entry on oeis.org

1, -1, -3, -3, -1, 10, 20, 36, 28, -11, -103, -245, -397, -448, -214, 464, 1817, 3680, 5660, 6473, 4362, -3232, -18428, -41946, -70589, -94890, -96996, -49673, 78907, 317995, 673299, 1105044, 1491333, 1605102, 1094914, -479358, -3561322, -8404118, -14781724, -21595744, -26450603, -25329527

Offset: 0

Ilya Gutkovskiy, Sep 15 2017

sign

Convolution inverse of A000294 (Euler transform of the triangular numbers).

Cf. A000294, A027999, A028377.

nmax = 41; CoefficientList[Series[Product[(1 - x^k)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

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

A327066 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^j).

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 17, 23, 41, 58, 93, 127, 205, 281, 423, 583, 869, 1180, 1716, 2322, 3317, 4479, 6282, 8406, 11696, 15589, 21343, 28325, 38480, 50756, 68307, 89688, 119725, 156586, 207449, 269921, 355530, 460804, 602816, 778281, 1012956, 1302481, 1686418

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A000294, A061256, A182269, A279216, A280540, A327063, A327067, A327068.

nmax = 50; CoefficientList[Series[Product[Product[1/(1-x^(k*j))^j, {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]

A327067 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^k).

Original entry on oeis.org

1, 1, 3, 6, 15, 26, 57, 101, 202, 358, 670, 1165, 2113, 3614, 6326, 10691, 18275, 30408, 50969, 83716, 137943, 223883, 363547, 583369, 935524, 1485673, 2355496, 3705275, 5815497, 9066696, 14100325, 21802824, 33622951, 51592978, 78949673, 120278899, 182742752

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A000294, A061256, A182269, A279216, A280540, A327064, A327066, A327068.

nmax = 40; CoefficientList[Series[Product[Product[1/(1-x^(k*j))^k, {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]

A327068 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(kj))^(kj)).

Original entry on oeis.org

1, 1, 3, 6, 17, 28, 66, 116, 248, 441, 867, 1516, 2894, 5015, 9138, 15724, 27954, 47428, 82421, 138380, 235910, 392040, 657590, 1081225, 1789550, 2914500, 4763562, 7689071, 12433581, 19897139, 31862226, 50583981, 80285138, 126509709, 199167763, 311620226

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A000294, A061256, A182269, A279216, A280540, A327065, A327066, A327067.

nmax = 40; CoefficientList[Series[Product[Product[1/(1-x^(k*j))^(k*j), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]

A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k-1)).

Original entry on oeis.org

1, 1, 7, 22, 71, 206, 616, 1712, 4743, 12677, 33407, 86085, 218677, 546060, 1345840, 3271893, 7861239, 18670881, 43883904, 102112483, 235401947, 537869136, 1218743007, 2739566083, 6111766043, 13536683750, 29775945929, 65065819486, 141285315728, 304935221675, 654318376244, 1396166024244, 2963068779402

Offset: 0

Ilya Gutkovskiy, Nov 28 2016

nonn

Euler transform of the hexagonal numbers (A000384).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000384, A000335, A023871.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(2*d-1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ exp(-Zeta'(-1) - Zeta(3)/(2*Pi^2) - 75*Zeta(3)^3/(4*Pi^8) - 15^(5/4)*Zeta(3)^2/(2^(9/4)*Pi^5) * n^(1/4) - sqrt(15/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(9/4)*Pi/(3^(5/4)*5^(1/4)) * n^(3/4)) / (2^(67/48) * 15^(5/48) * Pi^(1/12) * n^(29/48)). - Vaclav Kotesovec, Dec 02 2016

A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k-3)/2).

Original entry on oeis.org

1, 1, 8, 26, 88, 269, 843, 2456, 7115, 19892, 54756, 147355, 390517, 1017091, 2612670, 6617641, 16556913, 40933339, 100104289, 242276236, 580718077, 1379161494, 3247074738, 7581837910, 17564867853, 40388447308, 92206496318, 209069338580, 470944571003, 1054178579266, 2345477963043, 5188246121144, 11412352653001

Offset: 0

Ilya Gutkovskiy, Nov 28 2016

nonn

Euler transform of the heptagonal numbers (A000566).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000566, A000335, A023871.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(5*d-3)/2, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ exp(-3*Zeta'(-1)/2 - 5*Zeta(3)/(8*Pi^2) - 81*Zeta(3)^3/(2*Pi^8) - 3^(13/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - 3^(3/2)*Zeta(3)/(sqrt(2)*Pi^2) * sqrt(n) + 2^(7/4)*Pi/3^(5/4) * n^(3/4)) / (2^(51/32) * 3^(3/32) * Pi^(1/8) * n^(19/32)). - Vaclav Kotesovec, Dec 02 2016

A294654 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(5k-3)/2)(1 - x^(2k))^(k(5*k+3)/2)).

Original entry on oeis.org

1, 1, 5, 12, 35, 81, 208, 475, 1123, 2505, 5617, 12192, 26368, 55797, 117255, 242660, 498126, 1010515, 2033662, 4053214, 8017622, 15729219, 30643069, 59268267, 113898873, 217480476, 412813600, 779042099, 1462188257, 2729852845, 5070966794, 9373909586, 17247473718

Offset: 0

Ilya Gutkovskiy, Nov 06 2017

nonn

Euler transform of the generalized heptagonal numbers (A085787).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A000294, A085787, A278769, A294591, A294621, A294655.

nmax = 32; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (5 k - 3)/2) (1 - x^(2 k))^(k (5 k + 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (5 d (d + 1)/8 + (-1)^d (2 d + 1)/16 - 1/16), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A085787(k).

a(n) ~ exp(Pi * (2/3)^(5/4) * n^(3/4) + 5*Zeta(3) * sqrt(3*n) / (2^(3/2) * Pi^2) - (75*3^(1/4) * Zeta(3)^2 / (2^(13/4) * Pi^5) + Pi / (2^(17/4) * 3^(3/4))) * n^(1/4) + 375 * Zeta(3)^3 / (8*Pi^8) - 5*Zeta(3) / (64*Pi^2) + 1/12) * Pi^(1/12) / (A * 2^(11/6) * 3^(7/48) * n^(31/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 07 2017

cp's OEIS Frontend

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

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A264923 G.f.: 1 / Product_{n>=0} (1 - x^(n+3))^((n+1)*(n+2)/2!).

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A278403 a(n) = Sum_{d|n} d^2 * (d+1)/2.

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A292386 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)/2).

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A327066 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^j).

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A327067 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^k).

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A327068 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))^(k*j)).

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A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).

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A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

A327068 Expansion of Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(kj))^(kj)).

A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k-1)).

A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k-3)/2).

A294654 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(5k-3)/2)(1 - x^(2k))^(k(5*k+3)/2)).