A022597 -id:A022597 - cp's OEIS frontend

A089814 Expansion of Product_{k>=1} (1 - q^(10k-5))^2.

1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 5, 0, 0, 0, 0, -6, 0, 0, 0, 0, 9, 0, 0, 0, 0, -12, 0, 0, 0, 0, 13, 0, 0, 0, 0, -16, 0, 0, 0, 0, 21, 0, 0, 0, 0, -26, 0, 0, 0, 0, 29, 0, 0, 0, 0, -36, 0, 0, 0, 0, 46, 0, 0, 0, 0, -54, 0, 0, 0, 0, 62, 0, 0, 0, 0, -74, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

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G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A022597 (expansion of Product_{m >= 1} (1 + q^m)^(-2)).

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^(10*k-5))^2: k in [1..20]]))); // G. C. Greubel, Oct 20 2018

A022597[n_] := SeriesCoefficient[ Product[1 + q^k, {k, n}]^-2, {q, 0, n}]; a[n_] := If[Mod[n, 5] != 0, 0, A022597[n/5]]; a[0] = 1; Table[a[n], {n, 0, 96}] (* Jean-François Alcover, Nov 12 2012, after Michael Somos *)

x='x+O(x^100); Vec(prod(k=1,20, (1-x^(10*k-5))^2)) \\ G. C. Greubel, Oct 20 2018

a(5*n) = A022597(n). a(n) = 0 unless n == 0 (mod 5). - Michael Somos, Jun 08 2012

A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.

Original entry on oeis.org

1, -5, 5, 10, -15, -1, -30, 50, 65, -95, -1, -170, 220, 300, -380, 0, -635, 820, 1025, -1310, 0, -2045, 2525, 3140, -3845, 2, -5780, 7070, 8565, -10405, -1, -15130, 18125, 21760, -25960, 0, -36820, 43780, 51785, -61290, 0, -85170, 100030, 117500, -137550, 0

Offset: 0

Seiichi Manyama, Apr 29 2017

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

			G.f. = 1 - 5*x + 5*x^2 + 10*x^3 - 15*x^4 - x^5 - 30*x^6 + 50*x^7 + 65*x^8 - 95*x^9 + ...
G.f. = q^-5 - 5*q + 5*q^7 + 10*q^13 - 15*q^19 - q^25 - 30*q^31 + 50*q^37 + 65*q^43 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

(Product_{k>0} (1 - x^k) / (1 - x^(m*k)))^m: A022597 (m=2), A199659 (m=3), A112143 (m=4), this sequence (m=5).

Cf. A285928.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^5 / QPochhammer[ x^5]^5, {x, 0, n}]; (* Michael Somos, Apr 29 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^5, n))}; /* Michael Somos, Apr 29 2017 */

a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A116073(k)*a(n-k) for n > 0.
Expansion of q^(5/6) * eta(q)^5 / eta(q^5)^5 in powers of q. - Michael Somos, Apr 29 2017
Expansion of f(-x)^5 / f(-x^5)^5 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Apr 29 2017
Euler transform of period 5 sequence [-5, -5, -5, -5, 0, ...]. - Michael Somos, Apr 29 2017

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, -2, 1, 0, 1, -2, -3, -2, 0, 1, -2, -3, 10, 4, 0, 1, -2, -3, 4, -4, -4, 0, 1, -2, -3, 4, 14, -20, 5, 0, 1, -2, -3, 4, 6, -8, 41, -6, 0, 1, -2, -3, 4, 6, 16, -46, 2, 9, 0, 1, -2, -3, 4, 6, 6, -30, 14, -111, -12, 0, 1, -2, -3, 4, 6, 6, 0, -58

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,   1,  1,  1, ...
   0, -2,  -2, -2, -2, ...
   0,  1,  -3, -3, -3, ...
   0, -2,  10,  4,  4, ...
   0,  4,  -4, 14,  6, ...
   0, -4, -20, -8, 16, ...

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

Columns k=0..1 give A000007, A022597.
Rows n=0 gives A000012.
Main diagonal gives A252650.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: this sequence (m=-2), A290217 (m=-1), A290216 (m=1), A293377 (m=2).

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A089814 Expansion of Product_{k>=1} (1 - q^(10k-5))^2.

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A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.

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A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.

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cp's OEIS Frontend

A089814 Expansion of Product_{k>=1} (1 - q^(10k-5))^2.

Views

Author

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.

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Comments

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Mathematica

PARI

Formula

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} j*x^(j*i))^2.

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A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.