A160525 -id:A160525 - cp's OEIS frontend

A160527 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 118.

1, 4, 14, 40, 105, 252, 574, 1237, 2568, 5138, 9988, 18893, 34937, 63238, 112370, 196244, 337477, 572024, 956956, 1581321, 2583637, 4176495, 6684820, 10599939, 16661401, 25972485, 40171474, 61672695, 94017765, 142368024, 214211760, 320350725, 476299978

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			G.f. = 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 252*x^5 + 574*x^6 + ...
G.f. = q^17 + 4*q^41 + 14*q^65 + 40*q^89 + 105*q^113 + 252*q^137 + 574*q^161 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. A071746, A213261.

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))^3 /(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

See Maple code in A160525 for formula.
G.f.: Product_{n >= 1} (1 - x^(7*n))^3/(1 - x^n)^4. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(50*n/21)) * 5 / (196*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017

A160528 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 118.

Original entry on oeis.org

1, 5, 20, 65, 190, 506, 1265, 2986, 6745, 14645, 30767, 62745, 124706, 242110, 460337, 858673, 1574140, 2839862, 5048435, 8852562, 15327290, 26224173, 44372688, 74301095, 123200079, 202394897, 329596348, 532299955, 852914900, 1356426196, 2141819621

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			G.f. = 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 506*x^5 + 1265*x^6 + ...
G.f. = q^23 + 5*q^47 + 20*q^71 + 65*q^95 + 190*q^119 + 506*q^143 + 1265*q^167 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. A002300, A160525, A160526, A160527.

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))^4 /(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

See Maple code in A160525 for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))^4/(1 - x^n)^5. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(62*n/21)) * sqrt(31) / (4*sqrt(3) * 7^(5/2) * n). - Vaclav Kotesovec, Nov 10 2017

A160539 Coefficients in the expansion of C/B^7, in Watson's notation of page 118.

Original entry on oeis.org

1, 7, 35, 140, 490, 1547, 4522, 12404, 32298, 80430, 192759, 446656, 1004598, 2199953, 4703104, 9836820, 20167210, 40593651, 80335164, 156503088, 300457906, 568992893, 1063818868, 1965178600, 3589328246, 6485976525, 11602141453, 20555544212, 36087448852

Offset: 0

N. J. A. Sloane, Nov 14 2009

nonn

Watson's C and B are (essentially) defined as C = prod(n>=1, 1-q^(7*n)) and B = prod(n>=1, 1-q^n). - Joerg Arndt, Jul 30 2011
From Petros Hadjicostas, Sep 23 2019: (Start)
In Section 5 of his paper, p. 118, Watson defines A = x^(1/7)*f(-x^(24/7)), B = x*f(-x^24), and C = x^7*f(-x^168), where f(-x^2) = Product_{n >= 1} (1 - x^(2*n)). Note that in different sections of the paper, the definitions of A, B, and C change.
Letting q = x^24, we get B = q^(1/24) * Product_{n >= 1} (1 - q^n), C = q^(7/24) * Product_{n >= 1} (1 - q^(7*n)), and C/B^7 = Product_{n >= 1} (1 - q^(7*n))/(1 -q^n)^7. This is the reason Joerg Arndt above omits the factor q^(1/24) in the definition of B and the factor q^(7/24) in the definition of C.
(End)

			1 + 7*x^24 + 35*x^48 + 140*x^72 + 490*x^96 + 1547*x^120 + 4522*x^144 + ... = 1 + 7*q + 35*q^2 + 140*q^3 + 490*q^4 + 1547*q^5 + ... with q = x^24.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see p. 118 (def. of A, B, C) and p. 125 (expansion of C/B^7).

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)

N=66; x='x+O('x^N);
gf=eta(x^7)/eta(x)^7;
Vec(gf) /* Joerg Arndt, Jul 30 2011 */

G.f.: E7/E1^7 where E1 = P(q), E7 = P(q^7), and P(q) = prod(n>=1, 1-q^n). - Joerg Arndt, Jul 30 2011
G.f.: exp(sum(n>=1, (sigma(7*n)-sigma(n))*x^n/n ) ). - Joerg Arndt, Jul 30 2011
See also Maple code in A160525 for formula.
a(n) ~ 2^(5/4) * exp(4*Pi*sqrt(2*n/7)) / (7^(9/4) * n^(9/4)). - Vaclav Kotesovec, Nov 10 2016

A160526 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 118.

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 221, 427, 804, 1461, 2596, 4497, 7652, 12767, 20984, 33958, 54255, 85580, 133520, 206066, 315010, 477083, 716494, 1067316, 1578102, 2316569, 3377965, 4894045, 7047970, 10091120, 14369439, 20354090, 28687663, 40239129, 56183879

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			G.f. = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + ...
G.f. = q^11 + 3*q^35 + 9*q^59 + 22*q^83 + 51*q^107 + 108*q^131 + 221*q^155 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. A160525, A160527, A160528.

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))^2 /(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

See Maple code in A160525 for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))^2/(1 - x^n)^3. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(38*n/21)) * sqrt(19) / (4*sqrt(3) * 7^(3/2) * n). - Vaclav Kotesovec, Nov 10 2017

A160534 Coefficients in the expansion of B^7/C, in Watson's notation of page 118.

Original entry on oeis.org

1, -7, 14, 7, -49, 21, 35, 42, -56, -119, 105, -70, 147, 147, -133, -168, -231, 252, -154, 315, 441, 7, -644, -574, 595, -679, 735, 574, 196, -406, -840, 840, -1470, 854, 1260, 21, -1617, -966, 1575, -1176, 1785, 1470, 35, -1974, -2058, 1533, -1988, 1932, 2387, -301, -2170, -2016, 3087, -2422

Offset: 0

N. J. A. Sloane, Nov 14 2009

sign

			1-7*x^24+14*x^48+7*x^72-49*x^96+21*x^120+35*x^144+42*x^168-...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 7)=0, -6, -7), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 07 2017

a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, #*If[Mod[#, 7]==0, -6, -7]&]* a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 13 2017, after Alois P. Heinz *)

See Maple code in A160525 for formula.

Euler transform of period 7 sequence [ -7, -7, -7, -7, -7, -7, -6, ...]. - Alois P. Heinz, Jan 07 2017

A160533 Coefficients in the expansion of C^5/B^6, in Watson's notation of page 118.

Original entry on oeis.org

1, 6, 27, 98, 315, 918, 2492, 6367, 15495, 36145, 81326, 177219, 375461, 775544, 1565870, 3096615, 6008917, 11458720, 21502964, 39754385, 72485518, 130464603, 231989748, 407847488, 709365160, 1221364655, 2082872680, 3519963776, 5897536697, 9800358525

Offset: 0

N. J. A. Sloane, Nov 14 2009

nonn

			G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + ...
G.f. = q^29 + 6*q^53 + 27*q^77 + 98*q^101 + 315*q^125 + 918*q^149 + 2492*q^173 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. A160525, A160526, A160527, A160528.

nn = 29; CoefficientList[Series[Product[(1 - x^(7 n))^5/(1 - x^n)^6, {n, nn}], {x, 0, nn}], x] (* Michael De Vlieger, Nov 06 2016 *)

See Maple code in A160525 for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))^5/(1 - x^n)^6. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(74*n/21)) * sqrt(37) / (1372*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017

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A160527 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 118.

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A160528 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 118.

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A160539 Coefficients in the expansion of C/B^7, in Watson's notation of page 118.

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A160526 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 118.

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A160534 Coefficients in the expansion of B^7/C, in Watson's notation of page 118.

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A160533 Coefficients in the expansion of C^5/B^6, in Watson's notation of page 118.

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