A097242 -id:A097242 - cp's OEIS frontend

A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.

1, 1, 3, 5, 9, 14, 24, 35, 55, 81, 120, 171, 248, 345, 486, 669, 920, 1246, 1690, 2256, 3014, 3984, 5253, 6870, 8970, 11618, 15022, 19306, 24745, 31557, 40154, 50845, 64244, 80850, 101501, 126982, 158514, 197218, 244865, 303143, 374497, 461435

Offset: 0

James Sellers, Apr 04 2000

Sum of products of multiplicities of odd parts in all partitions of n (if there are no odd parts in a partition then product of multiplicities is considered to be 1). - Vladeta Jovovic, Feb 16 2005

The sequence A077285 is the same but with multiplicities of all parts.

			1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 14*x^5 + 24*x^6 + 35*x^7 + 55*x^8 + ...
q^-1 + q^11 + 3*q^23 + 5*q^35 + 9*q^47 + 14*q^59 + 24*q^71 + 35*q^83 + ...
a(6) = 24 since the 5 partitions 6 = 5+1 = 4+2 = 3+2+1 = 2+2+2 each contribute 1, the 3 partitions 4+1+1 = 3+3 = 2+2+1+1 each contribute 2, the partition 3+1+1+1 contributes 3, the partition 2+1+1+1+1 contributes 4, and the partition 1+1+1+1+1+1 contributes 6 to give total 24 = 5*1 + 3*2 + 1*3 + 1*4 + 1*6. - _Michael Somos_, Mar 09 2011

George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See formula (18) on page 3944.

Cf. A077285, A052992, A097242, A247661, A247662.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
      +add(b(n-i*j, i-1)*`if`(irem(i, 2)=1, j, 1), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 16 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] * If[Mod[i, 2] == 1, j, 1], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
QP = QPochhammer; s = QP[q^4] * (QP[q^6]^2 / (QP[q] * QP[q^2] * QP[q^3] * QP[q^12])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))} /* Michael Somos, Mar 09 2011 */

Expansion of q^(1/12) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos, Mar 09 2011
Euler transform of period 12 sequence [ 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, ...]. - Michael Somos, Mar 09 2011
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(12*k - 10)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 2)))^(-1). [Andrews, p. 10, equation (5.9)]
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*sqrt(2)*n). - Vaclav Kotesovec, Nov 28 2015

A328789 Expansion of (chi(x^3) / chi(-x^2))^2 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 7, 6, 11, 14, 17, 22, 32, 34, 49, 60, 72, 90, 117, 132, 171, 206, 245, 298, 369, 422, 522, 620, 728, 868, 1043, 1198, 1439, 1688, 1962, 2304, 2717, 3114, 3668, 4258, 4909, 5698, 6627, 7566, 8788, 10112, 11574, 13310, 15317, 17410, 20010

Offset: 0

Michael Somos, Oct 27 2019

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A097242.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328795.

			G.f. = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 6*x^7 + 11*x^8 + ...
G.f. = q^-1 + 2*q^23 + 2*q^35 + 3*q^47 + 4*q^59 + 7*q^71 + 6*q^83 + ...

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A097242, A112206, A328795.

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] QPochhammer[ -x^3, x^6])^2, {x, 0, n}];

{a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^6 + A)^2)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

Expansion of q^(1/12) * (eta(q^4) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, ...].
G.f.: Product_{k>=1} (1 + x^(6*k - 3))^2 / (1 - x^(4*k - 2))^2.
a(n) = A112206(2*n).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019

A328800 Expansion of chi(-x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 2, -2, 0, 0, 1, -2, 0, 0, 2, -2, 0, 0, 3, -3, 0, 0, 3, -3, 0, 0, 3, -3, 0, 0, 5, -5, 0, 0, 4, -5, 0, 0, 6, -5, 0, 0, 7, -7, 0, 0, 7, -8, 0, 0, 8, -8, 0, 0, 11, -11, 0, 0, 10, -12, 0

Offset: 0

Michael Somos, Oct 27 2019

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A328797.
G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A328796.

			G.f. = 1 - x - x^5 + x^8 + x^12 - x^13 + x^16 - x^17 + x^20 + ...
G.f. = q^-1 - q^5 - q^29 + q^47 + q^71 - q^77 + q^95 - q^101 + ...

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A097242, A328796, A328797, A328802.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6], {x, 0, n}];

{a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};

Expansion of q^(1/6) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, ...].
G.f.: Product_{k>=1} (1 - x^(2*k-1)) * (1 + x^(6*k-3)).
a(n) = (-1)^n * A328802. a(4*n) = A097242(n). a(4*n + 1) = -A328796(n). a(4*n + 2) = a(4*n + 3) = 0.

A116664 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k parts that appear exactly once (n>=0, k>=0).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 4, 0, 2, 2, 2, 3, 4, 0, 1, 3, 4, 3, 0, 3, 7, 1, 1, 5, 4, 6, 0, 4, 10, 2, 2, 6, 7, 9, 0, 7, 12, 5, 3, 7, 13, 11, 0, 1, 8, 18, 7, 5, 0, 11, 15, 18, 1, 1, 10, 25, 11, 8, 0, 13, 23, 24, 2, 2, 15, 32, 16, 13, 0, 16, 33, 32, 5, 3, 18, 43, 24, 19, 0, 23, 40

Offset: 0

Emeric Deutsch, Feb 22 2006

Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0)=A097242(n). Sum(k*T(n,k), k>=0)=A116665(n).

			T(10,2) = 3 because the only partitions of 10 into odd parts and having exactly 2 parts that appear only once are [9,1],[7,3] and [5,3,1,1].
Triangle starts:
1;
0, 1;
1, 0;
1, 1;
1, 0, 1;
1, 2, 0;
2, 1, 1;
1, 4, 0;

Alois P. Heinz, Table of n, a(n) for n = 0..600, flattened

Cf. A000009, A097242, A116665.

g:=product(1+t*x^(2*j-1)+x^(2*(2*j-1))/(1-x^(2*j-1)),j=1..30): gser:=simplify(series(g,x=0,30)): P[0]:=1: for n from 1 to 25 do P[n]:=coeff(gser,x^n) od: for n from 0 to 25 do seq(coeff(P[n],t,j),j=0..floor(sqrt(n))) od; # yields sequence in triangular form, with one extra 0 in some rows
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       expand(add(b(n-i*j, i-2)*`if`(j=1, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, k), k=0..floor(sqrt(n))))
        (b(n, n-irem(n+1, 2))):
seq(T(n), n=0..25);  # Alois P. Heinz, Mar 16 2014

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[Sum[b[n-i*j, i-2]*If[j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, k], {k, 0, Floor[Sqrt[n]]}]][b[n, n-Mod[n+1, 2]]]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)

G.f.: product(1+tx^(2j-1)+x^(4j-2)/(1-x^(2j-1)), j=1..infinity).

A328802 Expansion of chi(x) * chi(-x^3) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 5, 5, 0, 0, 4, 5, 0, 0, 6, 5, 0, 0, 7, 7, 0, 0, 7, 8, 0, 0, 8, 8, 0, 0, 11, 11, 0, 0, 10, 12, 0, 0, 13, 12, 0, 0, 15

Offset: 0

Michael Somos, Oct 28 2019

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A328795.
G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A097242.

			G.f. = 1 + x + x^5 + x^8 + x^12 + x^13 + x^16 + x^17 + x^20 + ...
G.f. = q^-1 + q^5 + q^29 + q^47 + q^71 + q^77 + q^95 + q^101 + ...

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A097242, A328795, A328796, A328800.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6], {x, 0, n}];

{a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};

Expansion of q^(1/6) * (eta(q^2)^2 * eta(q^3)) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, ...].
G.f.: Product_{k>=1} (1 + x^(2*k-1)) * (1 - x^(6*k-3)).
a(n) = (-1)^n * A328800. a(4*n) = A097242(n). a(4*n + 1) = A328796(n). a(4*n + 2) = a(4*n + 3) = 0.

cp's OEIS Frontend

A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.

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A328789 Expansion of (chi(x^3) / chi(-x^2))^2 in powers of x where chi() is a Ramanujan theta function.

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A328800 Expansion of chi(-x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.

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A116664 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k parts that appear exactly once (n>=0, k>=0).

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Maple

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A328802 Expansion of chi(x) * chi(-x^3) in powers of x where chi() is a Ramanujan theta function.

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