A104502 -id:A104502 - cp's OEIS frontend

A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0

Offset: 0

Ilya Gutkovskiy, May 11 2017

A(n,k) is the number of partitions of n in which no parts are multiples of k.

A(n,k) is also the number of partitions of n into at most k-1 copies of each part.

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  1,  2,  2,  2,  2,  ...
  0,  2,  2,  3,  3,  3,  ...
  0,  2,  4,  4,  5,  5,  ...
  0,  3,  5,  6,  6,  7,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Partition Function b_k
Index entries for sequences related to partitions

Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.
Main diagonal gives A000041.
Mirror of A061198.
Cf. A061199, A210485.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(n$2, k-1)[1]:
seq(seq(A(n, 1+d-n), n=0..d), d=0..16);  # Alois P. Heinz, Oct 17 2018

Table[Function[k, SeriesCoefficient[Product[(1 - x^(i k))/(1 - x^i), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x^k, x^k]/QPochhammer[x, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

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

A121589 Series expansion of (eta(q^9) / eta(q))^3 in powers of q.

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 221, 429, 810, 1476, 2631, 4572, 7802, 13056, 21519, 34918, 55935, 88452, 138332, 213990, 327852, 497592, 748833, 1117692, 1655719, 2434938, 3556791, 5161808, 7445631, 10677096, 15226658, 21599469, 30485268, 42817788

Offset: 1

Michael Somos, Aug 09 2006

nonn

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q + 3*q^2 + 9*q^3 + 22*q^4 + 51*q^5 + 108*q^6 + 221*q^7 + 429*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

Cf. A104502, A131986.

N:= 100: # to get a(1)..a(N)
S:= series(q*Product(1-q^(9*k),k=1..N/9)/Product((1-q^k)^3, k=1..N),q,N+1):
seq(coeff(S,q,n),n=1..N); # Robert Israel, Nov 02 2017

nmax = 40; CoefficientList[Series[Product[((1-x^(9*k))/(1-x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
QP = QPochhammer; s = (QP[q^9]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^9] / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Nov 02 2017 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^9 + A) / eta(x + A))^3, n))};

Euler transform of period 9 sequence [3, 3, 3, 3, 3, 3, 3, 3, 0, ...].
G.f.: x * (Product_{k>0} (1 - x^(9*k) / (1 - x^k))^3.
Expansion of c(q^3) / (3 * b(q)) = (c(q) / (3 * b(q^3))^3 in powers of q where b(), c() are cubic AGM functions.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u^2 - v) - 2 * u * v * ( 3 * (u + v) + 13 * u * v ).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v * (1 + 9 * v + 27 * v^2) * (1 + 9 * u + 27 * u^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * u2 - (1 + 3 * (u1 + u2)) * (u3 + u6 + 9 * u3 * u6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A131986.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (27 * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
a(1) = 1, a(n) = (3/(n-1))*Sum_{k=1..n-1} A116607(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017
Convolution inverse of A131986. Convolution cube of A104502. - Michael Somos, Nov 02 2017

Second formula corrected by Vaclav Kotesovec, Sep 07 2015

A104501 Coefficients of the A-Dyson Mod 27 identity.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 172, 225, 288, 371, 470, 598, 751, 945, 1177, 1468, 1815, 2245, 2757, 3386, 4133, 5043, 6121, 7425, 8966, 10818, 13001, 15610, 18677, 22324, 26600, 31662, 37582, 44560, 52701, 62261, 73387, 86406

Offset: 0

Eric W. Weisstein, Mar 11 2005

nonn

			1 +q +2*q^2 +3*q^3 +5*q^4 +7*q^5 +11*q^6 +15*q^7 +22*q^8 +30*q^9 +...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities

Cf. A104502, A104503, A104504.

QP = QPochhammer; (QP[1/x^15, x^27]*QP[1/x^12, x^27]*QP[x^27])/((1-1/x^15)* (1-1/x^12)*QP[x]) + O[x]^50 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 18 2017 *)

{a(n)=local(m); if(n<0, 0, m=sqrtint(24*n+1); polcoeff( sum(k= -((m-1)\18), (m+1)\18, (-1)^k*x^((9*k^2-k)*3/2),x*O(x^n))/ eta(x+x*O(x^n)), n))} /* Michael Somos, Mar 15 2006 */

{a(n)=if(n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2* prod(j=1, k-1, (1-x^(3*j))/(1-x^(j+1))/(1-x^(2*j))/(1-x^(2*j+1)), 1+O(x^(n-k^2+1)))/(1-x)^2, 1), n))} /* Michael Somos, Mar 15 2006 */

{a(n) = local(A); if( n<0, 0, A = eta(x + x*O(x^n)) ; polcoeff( sum(k=0, n, (k%3==0) * polcoeff(A, k) * x^k) / A, n))} /* Michael Somos, Sep 29 2007 */

Expansion of f(-q^12,-q^15)/f(-q,-q^2) in powers of q where f() is Ramanujan's theta function.
Given A=A0+A1+A2+A3+A4 is the 5-section, then 0= A1^2*A4^2 +2*A2^2*A3^2 -A1*A3^3 -A4*A2^3 -A1*A2*A3*A4.
G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-12))(1-x^(27k-15))/(1-x^k).
G.f.: 1+ Sum_{k>0} x^k^2 ( Product_{j=1..k-1} 1-x^(3j) )/( (Product_{j=1..2k-1} (1-x^j)) (Product_{j=1..k}(1-x^j)) ).
A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - Michael Somos, Sep 29 2007

Edited by Michael Somos, Mar 15 2006

A104503 Coefficients of the C-Dyson Mod 27 identity.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 189, 241, 308, 389, 492, 615, 770, 956, 1187, 1463, 1802, 2207, 2701, 3288, 3999, 4842, 5857, 7056, 8491, 10183, 12197, 14564, 17369, 20658, 24539, 29075, 34408, 40627, 47912, 56385, 66277

Offset: 0

Eric W. Weisstein, Mar 11 2005

nonn

			1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 10*q^6 + 14*q^7 + 20*q^8 + 27*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities

Cf. A104501, A104502, A104504.

QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; a[n_]:= SeriesCoefficient[f[-q^12, -q^15]/f[-q, -q^2], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 08 2018 *)

{a(n)=local(m); if(n<0, 0, m=sqrtint(24*n+25); polcoeff( sum(k= -((m-5)\18), (m+5)\18, (-1)^k*x^((9*k^2-5*k)*3/2),x*O(x^n))/ eta(x+x*O(x^n)), n))} /* Michael Somos, Mar 15 2006 */

{a(n)=if(n<1, n==0, polcoeff( sum(k=0, sqrtint(n+1)-1, x^(k^2+2*k)* prod(j=1, k, (1-x^(3*j))/(1-x^j)/(1-x^(2*j+1))/(1-x^(2*j+2)), 1+O(x^(n-k^2-2*k+1)))/(1-x)/(1-x^2) ), n))} /* Michael Somos, Mar 15 2006 */

{a(n) = local(A); if( n<0, 0, n++; A = eta(x + x*O(x^n)) ; polcoeff( - sum(k=0, n, (k%3==1) * polcoeff(A, k) * x^k) / A, n))} /* Michael Somos, Sep 29 2007 */

Expansion of f(-q^6,-q^21)/f(-q,-q^2) in powers of q where f() is Ramanujan's theta function.
Given A=A0+A1+A2+A3+A4 is the 5-section, then 0= A0^2*A3^2 +2*A1^2*A2^2 -A0*A2^3 -A3*A1^3 -A0*A1*A2*A3.
G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-6))(1-x^(27k-21))/(1-x^k).
G.f.: Sum_{k>0} x^(k^2+2k) ( Product_{j=1..k} 1-x^(3j) )/ ( (Product_{j=1..2k+2} (1-x^j)) (Product_{j=1..k}(1-x^j)) ).
A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - Michael Somos, Sep 29 2007

A104504 Coefficients of the D-Dyson mod 27 identity.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 209, 259, 330, 407, 512, 628, 782, 955, 1179, 1432, 1755, 2122, 2583, 3109, 3762, 4510, 5427, 6480, 7760, 9231, 11004, 13043, 15485, 18293, 21634, 25475, 30021, 35245, 41396, 48459, 56740

Offset: 0

Eric W. Weisstein, Mar 11 2005

nonn

			1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 8*q^6 + 10*q^7 + 15*q^8 + 19*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities

Cf. A104501, A104502, A104503.

QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y];
a[n_] := SeriesCoefficient[f[-q^3, -q^24]/f[-q, -q^2], {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Apr 08 2018 *)

{a(n)=local(m); if(n<0, 0, m=sqrtint(24*n+49); polcoeff( sum(k= -((m-7)\18), (m+7)\18, (-1)^k*x^((9*k^2-7*k)*3/2),x*O(x^n))/ eta(x+x*O(x^n)), n))} /* Michael Somos, Mar 15 2006 */

{a(n)=if(n<1, n==0, polcoeff( sum(k=0, sqrtint(n+1)-1, x^(k^2+3*k)* prod(j=1, k, (1-x^(3*j))/(1-x^j)/(1-x^(2*j+1))/(1-x^(2*j+2)), 1+O(x^(n-k^2-2*k+1)))/(1-x)/(1-x^2) ), n))} /* Michael Somos, Mar 15 2006 */

{a(n) = local(A); if( n<0, 0, n+=2; A = eta(x + x*O(x^n)) ; polcoeff( - sum(k=0, n, (k%3==2) * polcoeff(A, k) * x^k) / A, n))} /* Michael Somos, Sep 29 2007 */

Expansion of f(-q^3,-q^24)/f(-q,-q^2) in powers of q where f() is Ramanujan's theta function.
Given A=A0+A1+A2+A3+A4 is the 5-section, then 0= 2*A0^2*A1^2 +A2^2*A4^2 -A2*A0^3 -A4*A1^3 -A0*A1*A2*A4.
G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-3))(1-x^(27k-24))/(1-x^k).
G.f.: Sum_{k>0} x^(k^2+3k) ( Product_{j=1..k} 1-x^(3j) )/ ( (Product_{j=1..2k+2} (1-x^j)) (Product_{j=1..k}(1-x^j)) ).
A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - Michael Somos, Sep 29 2007

Edited by Michael Somos, Mar 15 2006

A213598 Number of partitions of n in which no parts are multiples of 49.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173524

Offset: 0

Michael Somos, Jun 14 2012

nonn

For n<49 we have a(n)=A000041(n), for n>=49 a(n)!=A000041(n).

In Fricke page 401, he gives the expansion sigma(omega) = q^4 + q^6 + 2q^8 + 3q^10 + 5q^12 + 7q^14 + 11q^16 + 15q^18 + ... where q = exp( Pi i omega).

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 15*q^9 + 22*q^10 + ...

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 401. Eq. (49)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.

Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A035959 (m=5), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10), A092885 (m=25), this sequence (m=49).

Cf. A000041, A287926.

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 49, n, 49}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^49] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 13 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^49 + A) / eta(x + A), n))};

Expansion of q^(-2) * eta(q^49) / eta(q) in powers of q.
Euler transform of period 49 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].
Given g.f. A(x) then B(x) = x^2 * A(x) satisfies 0 = f(B(x), B(x^2),
B(x^4)) where f(u, v, w) = u * v * w * (1 - 7*v^2) - (v - w) * (u - v) * (v^2 - u*w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 1 / (7 f(t)) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(49*k)) / (1 - x^k).
a(n) ~ exp(4*Pi*sqrt(2*n)/7) / (2^(1/4) * 7^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
a(n) = (1/n)*Sum_{k=1..n} A287926(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Jun 16 2017

A320611 Number of parts in all partitions of n in which no part occurs more than eight times.

Original entry on oeis.org

1, 3, 6, 12, 20, 35, 54, 86, 119, 182, 254, 366, 499, 697, 934, 1274, 1681, 2237, 2918, 3830, 4927, 6377, 8128, 10388, 13120, 16600, 20796, 26076, 32419, 40318, 49798, 61494, 75464, 92582, 112990, 137800, 167284, 202912, 245128, 295870, 355813, 427472, 511926

Offset: 1

Alois P. Heinz, Oct 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Column k=8 of A210485.

Cf. A104502.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(4*i*(i+1) [0, l[1]*j]+l)(b(n-i*j, min(n-i*j, i-1))), j=0..min(n/i, 8))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);

Table[Length[Flatten[Select[IntegerPartitions[n], Max[Tally[#][[All, 2]]] <= 8 &]]], {n, 43}] (* Robert Price, Jul 31 2020 *)

a(n) ~ log(3) * exp(4*Pi*sqrt(n/3)/3) / (Pi * sqrt(2) * 3^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 18 2018

cp's OEIS Frontend

A286653 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).

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A121589 Series expansion of (eta(q^9) / eta(q))^3 in powers of q.

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A104501 Coefficients of the A-Dyson Mod 27 identity.

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A104503 Coefficients of the C-Dyson Mod 27 identity.

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A104504 Coefficients of the D-Dyson mod 27 identity.

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A213598 Number of partitions of n in which no parts are multiples of 49.

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