A162891 -id:A162891 - cp's OEIS frontend

A263401 Expansion of Product_{k>=1} (1 + x^k - x^(2*k)).

1, 1, 0, 2, 0, 1, 3, 1, 1, 2, 6, 1, 4, 2, 5, 10, 5, 4, 9, 7, 8, 21, 9, 13, 13, 19, 13, 27, 32, 23, 29, 33, 27, 45, 37, 45, 79, 49, 57, 68, 82, 67, 101, 83, 109, 155, 124, 113, 174, 148, 171, 196, 215, 198, 262, 310, 269, 330, 314, 342, 414, 430, 393, 536, 493

Offset: 0

Vaclav Kotesovec, Jan 03 2016

nonn

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

Cf. A000726, A002390, A055922, A100847, A109389, A162891, A266686.

nmax = 80; CoefficientList[Series[Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[3]] = -1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] - If[j < 2*k, 0, p[[j - 2*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ sqrt(log(phi)) * phi^sqrt(8*n) / (2^(3/4)*sqrt(Pi)*n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

Original entry on oeis.org

1, 1, 3, 12, 72, 480, 3780, 35280, 372960, 4263840, 54432000, 758419200, 11436163200, 185253868800, 3214699488000, 59172265152000, 1163830187520000, 24097823253504000, 525794940582912000, 12073276215576576000, 290883846352619520000, 7318777466097377280000

Offset: 0

Seiichi Manyama, Oct 01 2017

nonn

			Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
For n = 5,
    partition      |                         |
--------------------------------------------------------------------
     5             -> one 5                  -> 1/(1!)       (= 1  )
   = 4 + 1         -> one 4 and one 1        -> 1/(1!*1!)    (= 1  )
   = 3 + 2         -> one 3 and one 2        -> 1/(1!*1!)    (= 1  )
   = 3 + 1 + 1     -> one 3 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 2 + 2 + 1     -> two 2 and one 1        -> 1/(2!*1!)    (= 1/2)
--------------------------------------------------------------------
                                                sum             4
So a(5) = 5! * 4 = 480.
For n = 6,
    partition      |                         |
--------------------------------------------------------------------
     6             -> one 6                  -> 1/(1!)       (= 1  )
   = 5 + 1         -> one 5 and one 1        -> 1/(1!*1!)    (= 1  )
   = 4 + 2         -> one 4 and one 2        -> 1/(1!*1!)    (= 1  )
   = 4 + 1 + 1     -> one 4 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 3 + 3         -> two 3                  -> 1/(2!)       (= 1/2)
   = 3 + 2 + 1     -> one 3, one 2 and one 1 -> 1/(1!*1!*1!) (= 1  )
   = 2 + 2 + 1 + 1 -> two 2 and two 1        -> 1/(2!*2!)    (= 1/4)
--------------------------------------------------------------------
                                                sum            21/4
So a(6) = 6! * 21/4 = 3780.

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293135.

Cf. A000726, A162891, A263401, A293141.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)/j!, j=0..min(2, n/i))))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]/j!, {j, 0, Min[2, n/i]}]]];
a[n_] := n! b[n, n];
a /@ Range[0, 23] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n) - n) * n^(n+1/2) / (sqrt(5) * n^(3/4)), where c = -polylog(2, -1/2 - I/2) - polylog(2, -1/2 + I/2) = 0.9669456127221570300837545... Equivalently, c = -Sum_{k>=1} (-1)^k * cos(Pi*k/4) / (k^2 * 2^(k/2-1)). - Vaclav Kotesovec, Oct 01 2017

A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

Original entry on oeis.org

1, 1, 4, 10, 29, 72, 200, 510, 1364, 3546, 9348, 24400, 64090, 167562, 439200, 1149360, 3010349, 7879832, 20633304, 54014950, 141422328, 370239300, 969323000, 2537696160, 6643839400, 17393731933, 45537549048, 119218684970, 312119004990, 817137724392, 2139295489200, 5600747143950

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

nonn

Euler transform of A032198.

Alois P. Heinz, Table of n, a(n) for n = 0..2392
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 13.

Cf. A006951, A032198, A088305, A162891, A258210, A329157, A329163, A386706, A387017.

Convolution inverse of A329157.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(a(j)*b(n-j$2), j=0..n-1))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Jul 25 2025

nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1 / (1 - x^k / (1 - x^k)^2).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
G.f.: Product_{k>=1} 1 / (1 - x^k)^A032198(k).
G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A088305.
a(n) ~ phi^(2*n-1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 07 2019
a(2^k) = A002878(2^k-1) for all nonnegative integers k. Follows from Cor. 4.5 on page 11 of Kassel-Reutenauer paper. - Michael De Vlieger, Jul 28 2025

A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).

Original entry on oeis.org

1, -1, 1, -3, 5, -8, 12, -21, 37, -59, 92, -153, 256, -409, 654, -1073, 1754, -2824, 4552, -7394, 12010, -19406, 31337, -50782, 82306, -133072, 215152, -348346, 563939, -912217, 1475604, -2388075, 3864808, -6252750, 10115987, -16369340, 26488326, -42857128

Offset: 0

Vaclav Kotesovec, Nov 16 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A003105, A162891.

Cf. A000726, A109389, A263401.

nmax = 50; CoefficientList[Series[1/Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ -p / (sqrt(5) * r^(n+1)), where r = -(sqrt(5)-1)/2 and p = Product_{n>1} 1/(1 + r^n - r^(2*n)) = 1.0964214808924344474065093...

A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).

Original entry on oeis.org

1, 2, 1, 6, 3, 6, 16, 12, 16, 22, 51, 36, 60, 62, 91, 154, 148, 176, 236, 278, 328, 552, 508, 670, 771, 988, 1068, 1438, 1844, 1998, 2401, 2882, 3300, 4030, 4640, 5406, 7212, 7584, 9072, 10480, 12612, 13964, 17024, 18860, 22545, 27298, 30340, 34372, 41068

Offset: 0

Vaclav Kotesovec, Oct 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A162891, A263401, A276527, A293138.

N:= 100:
P:= mul(1+2*x^m- x^(2*m), m=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,n), n=0..N); # Robert Israel, Oct 01 2017

nmax = 100; CoefficientList[Series[Product[1+2*x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2^(3/2) * sqrt(Pi) * n^(3/4)), where c = Pi^2/6 + log(1+sqrt(2))^2/2 + polylog(2, 3-2*sqrt(2))/2 - 2*polylog(2, sqrt(2)-1) = 1.18805291660775259061867850175092520191179528961165451864292...

A309950 G.f.: Product_{j>=1} (1 + p(x^j)), where p(x) is the g.f. of A000040.

Original entry on oeis.org

1, 2, 5, 11, 22, 43, 78, 140, 238, 405, 665, 1077, 1710, 2685, 4140, 6336, 9551, 14280, 21117, 30994, 45051, 65046, 93170, 132600, 187439, 263449, 367999, 511409, 706833, 972257, 1330929, 1813846, 2461090, 3325803, 4476276, 6002036, 8018216, 10674307, 14161656

Offset: 0

Alois P. Heinz, Aug 24 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000040, A006171, A008578, A162891, A309955.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, ithprime(n),
      add(b(j, 1)*(t-> b(t, min(t, i-1)))(n-i*j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i==1,
   Prime[n], Sum[b[j, 1]*Function[t,
   b[t, Min[t, i-1]]][n-i*j], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Sep 15 2022, after Alois P. Heinz *)

A319668 Expansion of Product_{k>=1} (1 - x^k - x^(2*k)).

Original entry on oeis.org

1, -1, -2, 0, 0, 3, 1, 3, 1, -2, 0, -3, -6, -4, 1, -8, 1, 2, 5, 5, 4, 9, 13, 7, 3, 1, 3, 7, -16, -9, -17, -13, -21, -5, -25, -33, -3, -3, -9, 22, -6, 11, 29, 29, 57, 37, 40, 31, 58, 18, 35, 40, 37, -24, -36, -34, -29, -60, -54, -98, -74, -124, -113, -156, -71, -35, -140, -46, -16, -61, -25

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

sign

Cf. A000010, A000358, A000726, A109389, A137569, A162891, A263401.

a:=series(mul((1-x^k-x^(2*k)),k=1..100),x=0,71): seq(coeff(a,x,n),n=0..70); # Paolo P. Lava, Apr 02 2019

nmax = 70; CoefficientList[Series[Product[(1 - x^k - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[Sum[EulerPhi[d/j] (Fibonacci[j - 1] + Fibonacci[j + 1]), {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 70}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k)))/(j*k)), where phi = Euler totient function (A000010).

A309733 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x^(2*k))).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 36, 62, 104, 174, 286, 478, 780, 1284, 2100, 3433, 5586, 9114, 14798, 24064, 39050, 63376, 102726, 166584, 269835, 437190, 707964, 1146480, 1855966, 3004748, 4863306, 7871798, 12739576, 20617652, 33364524, 53992834, 87369548, 141379728, 228769842

Offset: 0

Seiichi Manyama, Sep 22 2019

nonn

Cf. A006951, A055922, A162891.

nmax = 40; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 22 2019 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k/(1-x^(2*k))))

a(n) ~ phi^(n+1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 22 2019

A320286 Expansion of Product_{k>=1} 1/(1 - x^k - x^(2k) - x^(3k)).

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 51, 93, 184, 343, 654, 1211, 2286, 4217, 7865, 14521, 26912, 49600, 91669, 168800, 311305, 573058, 1055576, 1942437, 3575840, 6578762, 12106121, 22270404, 40972700, 75367724, 138644224, 255020102, 469095029, 862827347, 1587061299, 2919111935, 5369224903

Offset: 0

Ilya Gutkovskiy, Oct 09 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000010, A001935, A082303, A093305, A162891.

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/( &*[(1-x^k-x^(2*k)-x^(3*k)): k in [1..m+2]]))); // G. C. Greubel, Oct 24 2018

seq(coeff(series(mul(((1-x^k-x^(2*k)-x^(3*k)))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 25 2018

nmax = 36; CoefficientList[Series[Product[1/(1 - x^k - x^(2 k) - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Exp[-Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k) + x^(2 j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]

m=40; x='x+O('x^m); Vec(1/prod(k=1, m+2, (1-x^k-x^(2*k)-x^(3*k)))) \\ G. C. Greubel, Oct 24 2018

G.f.: exp(-Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k) + x^(2*j*k)))/(j*k)), where phi = Euler totient function (A000010).
From Vaclav Kotesovec, Oct 09 2018: (Start)
a(n) ~ s*p / r^(n+1), where
r = A192918 = ((17 + 3*sqrt(33))^(1/3) - 2/(17 + 3*sqrt(33))^(1/3) - 1)/3 = 0.54368901269207636157085597180174798652520329765098393524... is the real root of the equation 1 - r - r^2 - r^3 = 0,
s = (51 + 9*sqrt(33))/(4*(17 + 3*sqrt(33))^(1/3) + (17 + 3*sqrt(33))^(5/3) - 34 - 6*sqrt(33)) = 0.3362281169949410942253629540143324151579260900204592... is the real root of the equation -1 - 2*s + 44*s^3 = 0,
p = Product_{k>=2} 1/(1 - r^k - r^(2*k) - r^(3*k)) = 2.577933056783997593784130068093034525002002622982961271582417329674...
(End)

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A263401 Expansion of Product_{k>=1} (1 + x^k - x^(2*k)).

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A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

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A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

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A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).

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A293182 Expansion of Product_{k>=1} (1 + 2*x^k - x^(2*k)).

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A309950 G.f.: Product_{j>=1} (1 + p(x^j)), where p(x) is the g.f. of A000040.

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A319668 Expansion of Product_{k>=1} (1 - x^k - x^(2*k)).

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A309733 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x^(2*k))).

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A320286 Expansion of Product_{k>=1} 1/(1 - x^k - x^(2*k) - x^(3*k)).

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A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).

A320286 Expansion of Product_{k>=1} 1/(1 - x^k - x^(2k) - x^(3k)).