A081362 -id:A081362 - cp's OEIS frontend

A022601 Expansion of Product_{m>=1} (1+q^m)^(-6).

1, -6, 15, -26, 51, -102, 172, -276, 453, -728, 1128, -1698, 2539, -3780, 5505, -7882, 11238, -15918, 22259, -30810, 42438, -58110, 78909, -106392, 142770, -190698, 253179, -334266, 439581, -575784, 750613, -974316, 1260336, -1624702, 2086530, -2670162

Offset: 0

N. J. A. Sloane

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McKay-Thompson series of class 8F for the Monster group.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 51*x^4 - 102*x^5 + 172*x^6 - 276*x^7 + ...
T8F = 1/q - 6*q^3 + 15*q^7 - 26*q^11 + 51*q^15 - 102*q^19 + 172*q^23 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
Cf. A022571, A058088, A081362, A112145, A112150.
Column k=6 of A286352.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^6, {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^6, n))}; /* Michael Somos, Jul 01 2014 */

Expansion of chi(-x)^6 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 01 2014
Expansion of q^(1/4) * 2 * k'(q) / k(q)^(1/2) in powers of q where k() is the elliptic modulus. - Michael Somos, Jul 01 2014
Expansion of q^(1/4) * (eta(q) / eta(q^2))^6 in powers of q. - Michael Somos, Jul 01 2014
Euler transform of period 2 sequence [ -6, 0, ...]. - Michael Somos, Jul 01 2014
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (21 + 6*u*v). - Michael Somos, Jul 01 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022571. - Michael Somos, Jul 01 2014
Convolution inverse of A022571. Convolution sixth power of A081362. - Michael Somos, Jul 01 2014
a(n) = (-1)^n * A112150(n) = A058088(2*n) = A112145(2*n). - Michael Somos, Jul 01 2014
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-6*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A104575 Alternating sum of diagonals in A060177.

Original entry on oeis.org

1, -1, -2, -1, -1, 3, 1, 7, 4, 4, 4, 2, -9, -7, -7, -28, -17, -25, -15, -24, -11, -8, 34, 19, 53, 46, 108, 110, 106, 113, 122, 108, 75, 103, -16, -87, -107, -169, -329, -257, -574, -501, -676, -609, -749, -588, -808, -548, -521, -315, -240, 369, 485, 865, 1099, 1738, 2129, 2686, 3088, 3460, 4103, 4011, 4480, 3983

Offset: 0

Vladeta Jovovic, Apr 21 2005

A090794(n) = (A000041(n)-a(n))/2. A092306(n) = (A000041(n)+a(n))/2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A006951.

Cf. A000041, A008965, A081362, A092306, A268498.

CoefficientList[Series[Product[(1-2x^k)/(1-x^k),{k,70}],{x,0,70}],x] (* Harvey P. Dale, Jan 21 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1-x^k))) \\ Seiichi Manyama, Oct 05 2019

G.f.: Product_{i>0} (1 - 2*x^i)/(1 - x^i).

Euler transform of -A008965(n).

a(0)=1 prepended by Seiichi Manyama, Oct 05 2019

A246581 G.f.: x^((k^2 + k)/2) / (Product_{i=1..k} (1 - x^i) * Product_{r>=1} (1 + x^r)) with k = 2.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, -1, 1, -2, 1, -3, 2, -4, 3, -5, 5, -6, 7, -8, 10, -10, 13, -13, 17, -17, 21, -22, 27, -28, 33, -36, 41, -45, 50, -56, 62, -69, 75, -85, 92, -103, 111, -125, 135, -150, 162, -180, 195, -215, 232, -256, 278, -303, 329, -359, 390, -423

Offset: 0

N. J. A. Sloane, Aug 31 2014

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Empirical: 2*(-1)^n*a(n+1) is equal to the number of partitions mu of n such that the diagram of mu and the diagram of the transpose of mu have exactly n-1 cells in common (see below example). - John M. Campbell, Feb 01 2016

			From _John M. Campbell_, Feb 01 2016: (Start)
For example, letting n=9, there are 2*(-1)^n*a(n+1) = (-2)*(-3) = 6 partitions mu of n=9 such that the diagram of mu and the diagram of the transpose of mu have exactly n-1 cells in common: (5,2,1,1), (4,3,2), (4,3,1,1), (4,2,2,1), (4,2,1,1,1), (3,3,2,1). For example, the diagram of (3,3,2,1) is
   ooo
   ooo
   oo
   o
and the diagram of the transpose of (3,3,2,1) is
   oooo
   ooo
   oo
and these diagrams share exactly (n-1)=8 cells in common, when the diagrams are positioned so that the upper-left corners of both diagrams coincide. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc., 39 (No. 1, 2002), 51-85, MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=2.

For k=0 and 1 we get A081362, A027349 (apart from signs).

Cf. A218907, A246582, A246583.

fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(2);

k = 2; CoefficientList[Series[x^((k^2 + k)/2)/(Product[1 - x^i, {i, k}] Product[1 + x^r, {r, 1000}]), {x, 0, 56}], x] (* Michael De Vlieger, Feb 01 2016 *)

G.f.: x^3/((1-x)*(1-x^2)) * Product_{k>=1} 1/(1+x^k). - Vaclav Kotesovec, Mar 12 2016

a(n) ~ (-1)^(n+1) * 3^(1/4) * exp(sqrt(n/6)*Pi) / (2^(9/4)*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 12 2016

A284315 Expansion of Product_{k>=0} (1 - x^(3*k+2)) in powers of x.

Original entry on oeis.org

1, 0, -1, 0, 0, -1, 0, 1, -1, 0, 1, -1, 0, 2, -1, -1, 2, -1, -1, 3, -1, -2, 3, -1, -3, 4, 0, -4, 4, 0, -5, 5, 1, -7, 5, 2, -8, 6, 4, -10, 5, 5, -12, 6, 8, -14, 5, 10, -16, 5, 14, -19, 3, 17, -21, 2, 22, -23, -1, 26, -26, -3, 33, -28, -8, 38, -30, -12, 46, -32, -19

Offset: 0

Seiichi Manyama, Mar 25 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), this sequence (m=3), A284316 (m=4), A284317 (m=5).

Cf. A078182, A262928.

CoefficientList[Series[Product[1 - x^(3k + 2), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(3*k+2)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n) * Sum_{k=1..n} A078182(k) * a(n-k), a(0) = 1.

A284897 Expansion of Product_{k>=1} 1/(1+x^k)^(k^3) in powers of x.

Original entry on oeis.org

1, -1, -7, -20, -8, 99, 455, 958, 715, -3606, -17450, -44157, -61852, 19546, 419786, 1442212, 3084950, 3756436, -2155907, -27112107, -88277693, -187777531, -251308697, -5153980, 1182558343, 4299818445, 9988792754, 16075200671, 12020651310, -29802956283

Offset: 0

Seiichi Manyama, Apr 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..7180

Cf. A248882.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), this sequence (m=3), A284898 (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^3) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

Original entry on oeis.org

1, -1, -15, -66, -54, 725, 4580, 12739, 3346, -149076, -791226, -2182124, -1656973, 16553206, 100646954, 318795473, 506196578, -818806580, -9148048880, -36415709566, -87180585636, -70923559814, 484810027389, 2992082912770, 9866919438716, 19936695359140

Offset: 0

Seiichi Manyama, Apr 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..3995

Cf. A248883.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), this sequence (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^4) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^4))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

Original entry on oeis.org

1, -1, -31, -212, -284, 4935, 43719, 160002, -96747, -4914512, -31358932, -94515285, 97642670, 2823746182, 16834776254, 51617810512, -11233909783, -1137004349695, -7267899354808, -25263858110877, -24537905293857, 319397811973578, 2523465326904492

Offset: 0

Seiichi Manyama, Apr 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..2647

Cf. A248884.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), A284898 (m=4), this sequence (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^5) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^5))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284927(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A303131 Expansion of Product_{n>=1} (1 + (16*x)^n)^(-1/4).

Original entry on oeis.org

1, -4, -24, -1248, 1632, -267136, -669440, -56925184, 597165568, -19934894080, 61831327744, -3209599664128, 47593545383936, -840449808072704, 8113679782510592, -350055154021040128, 5703847053344768000, -57129722970675609600, 704939718429511778304

Offset: 0

Seiichi Manyama, Apr 19 2018

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This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/4, g(n) = -16^n.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), A303130 (b=3), this sequence (b=4), A303132 (b=5).

Cf. A303124, A303135.

CoefficientList[Series[(2/QPochhammer[-1, 16*x])^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)

a(n) ~ (-1)^n * exp(Pi*sqrt(n/24)) * 2^(4*n - 9/4) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A303132 Expansion of Product_{n>=1} (1 + (25*x)^n)^(-1/5).

Original entry on oeis.org

1, -5, -50, -3875, 2500, -2046250, -12409375, -1087687500, 13232343750, -907225000000, 1545669140625, -362705679687500, 6007095839843750, -224713698632812500, 2118331116210937500, -226812683210205078125, 4765872641563720703125

Offset: 0

Seiichi Manyama, Apr 19 2018

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This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/5, g(n) = -25^n.

In general, for h>=1, if g.f. = Product_{k>=1} (1 + (h^2*x)^k)^(-1/h), then a(n) ~ (-1)^n * exp(Pi*sqrt(n/(6*h))) * h^(2*n) / (2^(7/4) * 3^(1/4) * h^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

Seiichi Manyama, Table of n, a(n) for n = 0..500

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), A303130 (b=3), A303131 (b=4), this sequence (b=5).

Cf. A303125, A303136.

CoefficientList[Series[(2/QPochhammer[-1, 25*x])^(1/5), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2018 *)

a(n) ~ (-1)^n * exp(Pi*sqrt(n/30)) * 5^(2*n - 1/4) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 20 2018

A318582 Expansion of 1/(1 + x*Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, 0, 1, 0, 0, 0, 1, -1, 0, 1, -3, 2, -1, -3, 4, -4, 0, 3, -5, 4, 0, -2, 4, -1, 1, 0, 3, -2, 0, 6, -11, 9, -1, -13, 18, -17, 1, 13, -23, 17, -4, -8, 13, -8, 7, -6, 15, -10, -3, 33, -50, 42, 0, -56, 85, -72, 6, 59, -100, 75, -23, -34, 53, -44, 35

Offset: 0

Ilya Gutkovskiy, Aug 29 2018

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			G.f. = 1 - x - x^4 + x^5 - x^6 + x^8 - x^9 + x^10 + x^13 + x^17 - x^18 + x^20 - 3*x^21 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A331484.

Cf. A000009, A081362.

a:=series(1/(1+x*mul(1+x^k,k=1..100)),x=0,76): seq(coeff(a,x,n),n=0..75); # Paolo P. Lava, Apr 02 2019

nmax = 75; CoefficientList[Series[1/(1 + x Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

G.f.: 1/(1 + x*Sum_{k>=0} A000009(k)*x^k).

a(0) = 1; a(n) = -Sum_{k=1..n} A000009(k-1)*a(n-k).

cp's OEIS Frontend

A022601 Expansion of Product_{m>=1} (1+q^m)^(-6).

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A104575 Alternating sum of diagonals in A060177.

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A246581 G.f.: x^((k^2 + k)/2) / (Product_{i=1..k} (1 - x^i) * Product_{r>=1} (1 + x^r)) with k = 2.

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A284315 Expansion of Product_{k>=0} (1 - x^(3*k+2)) in powers of x.

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A284897 Expansion of Product_{k>=1} 1/(1+x^k)^(k^3) in powers of x.

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A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

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A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

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A303131 Expansion of Product_{n>=1} (1 + (16*x)^n)^(-1/4).

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