A341714 -id:A341714 - 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).

A355358 Coefficients in the expansion of A(x) = 1 / Product_{n>=0} (1 - x^(13n+1))(1 - x^(13n+3))(1 - x^(13n+4))(1 - x^(13n+9))(1 - x^(13n+10))(1 - x^(13*n+12)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 5, 6, 8, 10, 11, 15, 18, 21, 25, 31, 36, 43, 50, 59, 69, 81, 93, 109, 126, 146, 168, 194, 222, 256, 291, 333, 379, 432, 489, 557, 629, 712, 805, 909, 1021, 1152, 1293, 1452, 1627, 1824, 2037, 2281, 2544, 2838, 3162, 3525, 3916, 4356

Offset: 0

Paul D. Hanna, Jul 31 2022

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 11*x^11 + 15*x^12 + 18*x^13 + 21*x^14 + ...
and the related series B(x) begins
B(x) = 1 + x^2 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + 4*x^11 + 6*x^12 + 6*x^13 + 8*x^14 + 9*x^15 + ... + A355359(n)*x^n + ...
such that A(x) and B(x) satisfy
1 = A(x^3)*B(x) - x^2*A(x)*B(x^3),
and
1 = A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2.
Related expansions begin
A(x^3)*B(x) = 1 + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + 10*x^11 + 13*x^12 + 14*x^13 + 20*x^14 + 24*x^15 + ...
A(x)*B(x^3) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + 13*x^10 + ...
A(x)^3*B(x) = 1 + 3*x + 7*x^2 + 16*x^3 + 34*x^4 + 65*x^5 + 120*x^6 + 213*x^7 + 365*x^8 + 609*x^9 + 994*x^10 + ...
A(x)*B(x)^3 = 1 + x + 4*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 35*x^6 + 54*x^7 + 94*x^8 + 142*x^9 + 232*x^10 + ...
A(x)^2*B(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + 185*x^8 + 300*x^9 + 481*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Srinivasa Ramanujan, Algebraic relations between certain infinite products, Proceedings of the London Mathematical Society, vol.2, no.18, 1920.

Cf. A355359, A341714, A214157.

nmax = 60; CoefficientList[Series[1/Product[(1 - x^(13*n + 1))*(1 - x^(13*n + 3))*(1 - x^(13*n + 4))*(1 - x^(13*n + 9))*(1 - x^(13*n + 10))*(1 - x^(13*n + 12)), {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2022 *)

{a(n) = polcoeff( 1/prod(m=0, n, (1 - x^(13*m+1))*(1 - x^(13*m+3))*(1 - x^(13*m+4))*(1 - x^(13*m+9))*(1 - x^(13*m+10))*(1 - x^(13*m+12)), 1 + x*O(x^n)), n)};
for(n=0,60,print1(a(n),", "))

{a(n) = polcoeff( sqrt( prod(k=1, n, (1 - x^(13*k))/(1 - x^k)^(1 + kronecker(13, k)), 1 + x*O(x^n)) ), n)};
for(n=0,60,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n and related function B(x) satisfy the following relations.
(1.a) A(x^3)*B(x) - x^2*A(x)*B(x^3) = 1.
(1.b) A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2 = 1.
(2.a) A(x) = 1 / Product_{n>=0} (1 - x^(13*n+1))*(1 - x^(13*n+3))*(1 - x^(13*n+4))*(1 - x^(13*n+9))*(1 - x^(13*n+10))*(1 - x^(13*n+12)).
(2.b) B(x) = 1 / Product_{n>=0} (1 - x^(13*n+2))*(1 - x^(13*n+5))*(1 - x^(13*n+6))*(1 - x^(13*n+7))*(1 - x^(13*n+8))*(1 - x^(13*n+11)).
(3.a) A(x)*B(x) = Product_{n>=1} (1 - x^(13*n))/(1 - x^n), a g.f. of A341714.
(3.b) A(x)/B(x) = Product_{n>=1} 1/(1 - x^n)^Kronecker(13, n), a g.f. of A214157.
(4.a) A(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 + Kronecker(13, n)) ).
(4.b) B(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 - Kronecker(13, n)) ).
(5.a) A(x) = ( Product_{n>=1} (1 - x^(13*n))^3 ) / ( f(-x, -x^12) * f(-x^3, -x^10) * f(-x^4, -x^9) ).
(5.b) B(x) = ( Product_{n>=1} (1 - x^(13*n))^3 ) / ( f(-x^2, -x^11) * f(-x^5, -x^8) * f(-x^6, -x^7) ) .
Formulas (4.*) and (5.*) are derived from formulas given by Michael Somos in A214157, where f(a,b) = Sum_{n=-oo..+oo} a^(n*(n+1)/2) * b^(n*(n-1)/2) is Ramanujan's theta function..
a(n) ~ exp(2*Pi*sqrt(n/13)) / (16 * 13^(1/4) * sin(Pi/13) * sin(3*Pi/13) * cos(5*Pi/26) * n^(3/4)). - Vaclav Kotesovec, Aug 01 2022

A355359 Coefficients in the expansion of B(x) = 1 / Product_{n>=0} (1 - x^(13n+2))(1 - x^(13n+5))(1 - x^(13n+6))(1 - x^(13n+7))(1 - x^(13n+8))(1 - x^(13*n+11)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 3, 2, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 22, 24, 29, 32, 39, 43, 53, 57, 69, 75, 90, 99, 117, 129, 150, 166, 193, 213, 246, 273, 312, 346, 394, 436, 496, 549, 621, 687, 774, 855, 962, 1062, 1192, 1313, 1470, 1618, 1807, 1989, 2214, 2436

Offset: 0

Paul D. Hanna, Aug 01 2022

nonn

			G.f.: B(x) = 1 + x^2 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + 4*x^11 + 6*x^12 + 6*x^13 + 8*x^14 + 9*x^15 + ...
and the related series A(x) begins
A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 11*x^11 + 15*x^12 + 18*x^13 + 21*x^14 + ... + A355358(n)*x^n + ...
such that A(x) and B(x) satisfy
1 = A(x^3)*B(x) - x^2*A(x)*B(x^3),
and
1 = A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2.
Related expansions begin
A(x^3)*B(x) = 1 + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + 10*x^11 + 13*x^12 + 14*x^13 + 20*x^14 + 24*x^15 + ...
A(x)*B(x^3) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + 13*x^10 + ...
A(x)^3*B(x) = 1 + 3*x + 7*x^2 + 16*x^3 + 34*x^4 + 65*x^5 + 120*x^6 + 213*x^7 + 365*x^8 + 609*x^9 + 994*x^10 + ...
A(x)*B(x)^3 = 1 + x + 4*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 35*x^6 + 54*x^7 + 94*x^8 + 142*x^9 + 232*x^10 + ...
A(x)^2*B(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + 185*x^8 + 300*x^9 + 481*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Srinivasa Ramanujan, Algebraic relations between certain infinite products, Proceedings of the London Mathematical Society, vol.2, no.18, 1920.

Cf. A355358, A341714, A214157.

nmax = 60; CoefficientList[Series[1/Product[(1 - x^(13*n + 2))*(1 - x^(13*n + 5))*(1 - x^(13*n + 6))*(1 - x^(13*n + 7))*(1 - x^(13*n + 8))*(1 - x^(13*n + 11)), {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2022 *)

{a(n) = polcoeff( 1/prod(m=0, n, (1 - x^(13*m+2))*(1 - x^(13*m+5))*(1 - x^(13*m+6))*(1 - x^(13*m+7))*(1 - x^(13*m+8))*(1 - x^(13*m+11)), 1 + x*O(x^n)), n)};
for(n=0,60,print1(a(n),", "))

{a(n) = polcoeff( sqrt( prod(k=1, n, (1 - x^(13*k))/(1 - x^k)^(1 - kronecker(13, k)), 1 + x*O(x^n)) ), n)};
for(n=0,60,print1(a(n),", "))

G.f. B(x) = Sum_{n>=0} a(n)*x^n and related function A(x) satisfy the following relations.
(1.a) A(x^3)*B(x) - x^2*A(x)*B(x^3) = 1.
(1.b) A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2 = 1.
(2.a) A(x) = 1 / Product_{n>=0} (1 - x^(13*n+1))*(1 - x^(13*n+3))*(1 - x^(13*n+4))*(1 - x^(13*n+9))*(1 - x^(13*n+10))*(1 - x^(13*n+12)).
(2.b) B(x) = 1 / Product_{n>=0} (1 - x^(13*n+2))*(1 - x^(13*n+5))*(1 - x^(13*n+6))*(1 - x^(13*n+7))*(1 - x^(13*n+8))*(1 - x^(13*n+11)).
(3.a) A(x)*B(x) = Product_{n>=1} (1 - x^(13*n))/(1 - x^n), a g.f. of A341714.
(3.b) A(x)/B(x) = Product_{n>=1} 1/(1 - x^n)^Kronecker(13, n), a g.f. of A214157.
(4.a) A(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 + Kronecker(13, n)) ).
(4.b) B(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 - Kronecker(13, n)) ).
(5.a) A(x) = ( Product_{n>=1} (1 - x^(13*n))^3 ) / ( f(-x, -x^12) * f(-x^3, -x^10) * f(-x^4, -x^9) ).
(5.b) B(x) = ( Product_{n>=1} (1 - x^(13*n))^3 ) / ( f(-x^2, -x^11) * f(-x^5, -x^8) * f(-x^6, -x^7) ) .
Formulas (4.*) and (5.*) are derived from formulas given by Michael Somos in A214157, where f(a,b) = Sum_{n=-oo..+oo} a^(n*(n+1)/2) * b^(n*(n-1)/2) is Ramanujan's theta function..
a(n) ~ exp(2*Pi*sqrt(n/13)) / (16 * 13^(1/4) * sin(2*Pi/13) * cos(Pi/26) * cos(3*Pi/26) * n^(3/4)). - Vaclav Kotesovec, Aug 01 2022

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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A355358 Coefficients in the expansion of A(x) = 1 / Product_{n>=0} (1 - x^(13n+1))(1 - x^(13n+3))(1 - x^(13n+4))(1 - x^(13n+9))(1 - x^(13n+10))(1 - x^(13*n+12)).

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A355359 Coefficients in the expansion of B(x) = 1 / Product_{n>=0} (1 - x^(13n+2))(1 - x^(13n+5))(1 - x^(13n+6))(1 - x^(13n+7))(1 - x^(13n+8))(1 - x^(13*n+11)).

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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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Maple

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Formula

A355358 Coefficients in the expansion of A(x) = 1 / Product_{n>=0} (1 - x^(13*n+1))*(1 - x^(13*n+3))*(1 - x^(13*n+4))*(1 - x^(13*n+9))*(1 - x^(13*n+10))*(1 - x^(13*n+12)).

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Formula

A355359 Coefficients in the expansion of B(x) = 1 / Product_{n>=0} (1 - x^(13*n+2))*(1 - x^(13*n+5))*(1 - x^(13*n+6))*(1 - x^(13*n+7))*(1 - x^(13*n+8))*(1 - x^(13*n+11)).

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Examples

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Programs

Mathematica

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A355358 Coefficients in the expansion of A(x) = 1 / Product_{n>=0} (1 - x^(13n+1))(1 - x^(13n+3))(1 - x^(13n+4))(1 - x^(13n+9))(1 - x^(13n+10))(1 - x^(13*n+12)).

A355359 Coefficients in the expansion of B(x) = 1 / Product_{n>=0} (1 - x^(13n+2))(1 - x^(13n+5))(1 - x^(13n+6))(1 - x^(13n+7))(1 - x^(13n+8))(1 - x^(13*n+11)).