A255528 -id:A255528 - cp's OEIS frontend

A284993 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j^k) in powers of x.

1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 1, -1, 1, -1, -15, -20, 0, 0, 1, 1, -1, -31, -66, -8, 11, 4, -1, 1, -1, -63, -212, -54, 99, 42, 2, 2, 1, -1, -127, -666, -284, 725, 455, 63, 8, -2, 1, -1, -255, -2060, -1350, 4935, 4580, 958, 73

Offset: 0

Seiichi Manyama, Apr 07 2017

			Square array begins:
   1,  1,  1,   1,   1,    1, ...
  -1, -1, -1,  -1,  -1,   -1, ...
   0, -1, -3,  -7, -15,  -31, ...
  -1, -2, -6, -20, -66, -212, ...
   1,  1,  0,  -8, -54, -284, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give A081362, A255528, A284896, A284897, A284898, A284899.

Cf. A144048, A284992.

G.f. of column k: Product_{j>=1} 1/(1+x^j)^(j^k).

A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 9, 5, 17, 17, 38, 33, 88, 75, 169, 181, 343, 353, 712, 728, 1348, 1518, 2591, 2898, 5025, 5615, 9259, 10866, 17160, 20111, 31775, 37264, 57130, 68782, 102663, 123698, 183793, 221708, 323077, 395325, 566079, 693248, 987086, 1210110, 1700074, 2100674, 2915549

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Convolution of A192065 and A255528.

Weigh transform of A001065.

N. J. A. Sloane, Transforms

Cf. A000203, A001065, A192065, A255528, A318784.

with(numtheory): a:=series(mul((1+x^k)^(sigma(k)-k),k=1..100),x=0,47): seq(coeff(a,x,n),n=0..46); # Paolo P. Lava, Apr 02 2019

nmax = 46; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]

G.f.: Product_{k>=1} (1 + x^k)^A001065(k).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*(sigma_1(d) - d) ) * x^k/k).
a(n) ~ exp(-Pi^4 / (864*(Pi^2 - 6)*Zeta(3)) - Pi^2 * n^(1/3) / (12*(2*(Pi^2 - 6)*Zeta(3))^(1/3)) + 3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (2^(17/24) * sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018

A268501 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k))^k.

Original entry on oeis.org

1, 0, 2, 6, 11, 32, 60, 148, 279, 690, 1312, 2778, 5684, 11282, 22920, 44724, 87919, 168978, 329800, 623086, 1189794, 2235744, 4189442, 7795642, 14438670, 26577246, 48616050, 88724110, 160629612, 290267100, 521225220, 933031364, 1661954928, 2950946220

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A022629, A255528, A266891, A268500, A268502.

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

A305628 Expansion of Product_{k>=1} 1/(1 + x^k)^(k+1).

Original entry on oeis.org

1, -2, 0, -2, 5, -2, 7, -6, 11, -20, 13, -32, 31, -50, 60, -70, 124, -112, 192, -198, 295, -364, 422, -616, 661, -1002, 1034, -1500, 1737, -2208, 2808, -3234, 4462, -4876, 6735, -7464, 9990, -11610, 14410, -17866, 20947, -27082, 30493, -40056, 45147, -58196, 66999, -83278, 99641

Offset: 0

Ilya Gutkovskiy, Aug 11 2018

sign

Convolution of A081362 and A255528.

Convolution inverse of A219555.

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

Cf. A081362, A219555, A255528.

nmax = 48; CoefficientList[Series[Product[1/(1 + x^k)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[(-1)^k x^k (2 - x^k)/(k (1 - x^k)^2), {k, 1, nmax}]], {x,0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d (d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

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

a(n) ~ (-1)^n * exp(3 * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (3 * 2^(7/3) * Zeta(3)^(1/3)) - 1/12 - Pi^4 / (864 * Zeta(3))) * A * Zeta(3)^(5/36) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Aug 21 2018

A301831 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 + x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, 0, 0, 6, -16, 16, -34, 217, -681, 1343, -3466, 13370, -42380, 109477, -312448, 1040248, -3267138, 9447529, -28367596, 90504001, -283611105, 861087913, -2654231074, 8386506600, -26359974392, 81902319183, -256179313766, 809890745232, -2557697524240, 8046530976599

Offset: 0

Ilya Gutkovskiy, Mar 27 2018

sign

			G.f. A(x) = 1 - x + 6*x^4 - 16*x^5 + 16*x^6 - 34*x^7 + 217*x^8 - 681*x^9 + 1343*x^10 - 3466*x^11 + ...
log(A(x)) = -x - x^2/2 - x^3/3 + 23*x^4/4 - 51*x^5/5 + 35*x^6/6 - 197*x^7/7 + ... + A281266(n)*x^n/n + ...

Cf. A109085, A181315, A255528, A281266, A301455, A301456, A301624.

G.f. satisfies: A(x) = exp(Sum_{k>=1} (-1)^k*x^k*A(x)^k/(k*(1 - x^k*A(x)^k)^2)).

a(n) = [x^n] (Sum_{k>=0} A255528(k)*x^k)^(n+1)/(n + 1).

A305255 a(n) = [x^n] exp(Sum_{k>=1} (-1)^kx^k/(k(1 - x^k)^n)).

Original entry on oeis.org

1, -1, -1, -4, -3, 14, 240, 1686, 9479, 36761, 3412, -1951731, -27296124, -268495319, -2093667873, -11586874946, -3788945531, 1127535019748, 21900095232973, 297591401221473, 3270627818325128, 28116733997044842, 129815302615081267, -1568168714539146596, -59839621829784309343

Offset: 0

Ilya Gutkovskiy, May 28 2018

sign

N. J. A. Sloane, Transforms

Cf. A255528, A293554, A294846, A305205, A305206.

Table[SeriesCoefficient[Exp[Sum[(-1)^k x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[1/(1 + x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 24}]

a(n) = [x^n] Product_{k>=1} 1/(1 + x^k)^binomial(n+k-2,n-1).

A319109 Expansion of Product_{k>=1} 1/(1 + x^k)^(k-1).

Original entry on oeis.org

1, 0, -1, -2, -2, -2, 0, 2, 7, 8, 12, 10, 9, -2, -10, -32, -40, -62, -62, -70, -37, -20, 57, 106, 224, 272, 388, 376, 431, 272, 192, -184, -414, -1012, -1321, -2020, -2157, -2700, -2318, -2352, -1014, -272, 2280, 3798, 7464, 9200, 13257, 13958, 17098, 14846, 15266

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

sign

Convolution of A000009 and A255528.

Convolution inverse of A052812.

Cf. A000009, A052812, A255528, A305628.

a:=series(mul(1/(1+x^k)^(k-1),k=1..100),x=0,51): seq(coeff(a,x,n),n=0..50); # Paolo P. Lava, Apr 02 2019

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^k x^(2 k)/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d (d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

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

cp's OEIS Frontend

A284993 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j^k) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A268501 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k))^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A305628 Expansion of Product_{k>=1} 1/(1 + x^k)^(k+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A301831 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 + x^k*A(x)^k)^k.

Views

Author

Keywords

Examples

Crossrefs

Formula

A305255 a(n) = [x^n] exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A319109 Expansion of Product_{k>=1} 1/(1 + x^k)^(k-1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

A305255 a(n) = [x^n] exp(Sum_{k>=1} (-1)^kx^k/(k(1 - x^k)^n)).