A131942 -id:A131942 - cp's OEIS frontend

A130126 Number of partitions of n in which each even part has odd multiplicity.

1, 1, 2, 3, 4, 6, 10, 13, 17, 24, 33, 43, 58, 75, 98, 127, 161, 205, 262, 328, 414, 517, 641, 794, 982, 1205, 1475, 1803, 2197, 2664, 3230, 3896, 4693, 5640, 6754, 8077, 9647, 11479, 13637, 16178, 19152, 22624, 26695, 31426, 36948, 43372, 50819, 59463, 69490

Offset: 0

Vladeta Jovovic, Aug 01 2007

			a(5) = 6 because we have 5, 41, 32, 311, 2111 and 11111 (221 does not qualify).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A131942.

g:=product((1+q^(2*n)-q^(4*n))/((1-q^(2*n-1))*(1-q^(4*n))),n=1..50): gser:= series(g,q=0,45): seq(coeff(gser,q,n),n=0..42); # Emeric Deutsch, Aug 24 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      add(`if`(irem(i, 2)=0 and irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 27 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 0, 0, b[n - i*j, i - 1]], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k) - x^(4*k))/((1-x^(2*k-1)) * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{n>=1} (1+q^(2n)-q^(4n))/((1-q^(2n-1))(1-q^(4n))).

a(n) ~ sqrt(Pi^2/2 + 4*log(phi)^2) * exp(sqrt((Pi^2 + 8*log(phi)^2)*(n/2))) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

More terms from Emeric Deutsch, Aug 24 2007

A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

Original entry on oeis.org

1, 2, 3, 7, 10, 17, 28, 42, 62, 93, 137, 193, 276, 383, 532, 734, 997, 1342, 1807, 2400, 3177, 4190, 5478, 7130, 9245, 11923, 15305, 19591, 24957, 31673, 40075, 50518, 63460, 79523, 99296, 123664, 153616, 190271, 235072, 289776, 356302, 437107, 535112, 653626

Offset: 0

Vladeta Jovovic, Aug 16 2007

			a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A055922, A117958, A130126, A131942, A102247, A263401.

g:=product((1+x^i-x^(2*i))/(1-x^i),i=1..50): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..35); # Emeric Deutsch, Aug 25 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(2*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^i).

a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((2*Pi^2/3 + 8*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

More terms from Emeric Deutsch, Aug 25 2007

A131945 Number of partitions of n where odd parts are distinct or repeated once.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 45, 55, 74, 90, 119, 145, 188, 228, 291, 351, 442, 532, 664, 796, 982, 1172, 1435, 1708, 2076, 2462, 2972, 3512, 4214, 4966, 5929, 6965, 8272, 9688, 11457, 13383, 15762, 18362, 21543, 25031, 29264, 33922, 39533, 45717

Offset: 0

Brian Drake, Jul 30 2007

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

Also number of partitions of n such that every part is not congruent to 3 mod 6. More generally, g.f. for number of partitions of n such that every odd part occurs at most m times is product_{n=1..oo} (1-q^((m+1)*(2*n-1)))/(1-q^n). Similarly, g.f. for number of partitions of n such that every even part occurs at most m times is product_{n=1..oo} (1-q^((2*m+2)*n))/(1-q^n). - Vladeta Jovovic, Aug 01 2007

			a(6) = 8 because we have 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 2+2+2 and 2+2+1+1. The three excluded partitions of 6 are 3+1+1+1, 2+1+1+1+1 and 1+1+1+1+1+1.
G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...
G.f. = 1/q + q^5 + 2*q^11 + 2*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 10*q^41 + ...

Brian Drake and Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Mircea Merca, Overpartitions in terms of 2-adic valuation, Aequat. Math. (2024). See p. 11.
James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 2.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A006950, A010054, A121373, A131942, A279320.

A:= series(product( (1-q^(6*n-3))/(1-q^n), n=1..20),q,21): seq(coeff(A,q,i), i=0..20);

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
nmax = 50; CoefficientList[Series[Product[1 / ((1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 11 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Aug 05 2007 */

G.f.: product_{n=1..oo} (1-q^(6n-3))/(1-q^n).
Expansion of chi(-x^3) / f(-x) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Aug 05 2007
Expansion of q^(1/6) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Aug 05 2007
Euler transform of period 6 sequence [ 1, 1, 0, 1, 1, 1, ...]. - Michael Somos, Aug 05 2007
a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*n). - Vaclav Kotesovec, Dec 11 2016

A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

Original entry on oeis.org

1, 1, 0, 2, 2, 3, 2, 4, 7, 8, 8, 10, 17, 17, 20, 26, 39, 39, 46, 56, 77, 85, 96, 116, 154, 172, 190, 234, 289, 328, 364, 440, 532, 610, 670, 808, 957, 1091, 1204, 1432, 1675, 1905, 2110, 2476, 2867, 3255, 3608, 4184, 4837, 5451, 6050, 6960, 7980, 8961, 9972, 11370

Offset: 0

Vladeta Jovovic, Aug 16 2007

			a(7) = 4 because we have 7, 322, 22111 and 1111111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A055922, A117958, A130126, A131942, A100847.

g:=product((1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)),i=1..40): gser:=series(g,x=0, 60): seq(coeff(gser,x,n),n=0..55); # Emeric Deutsch, Aug 23 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, b(n-i*j, i-1), 0), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{i>0} (1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)).

a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((Pi^2/3 + 4*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

More terms from Emeric Deutsch, Aug 23 2007

A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 2, 6, 21, 105, 675, 4725, 35805, 322245, 3236625, 35602875, 425872755, 5536345815, 77347084815, 1160206272225, 18403556596425, 312860462139225, 5643104418376425, 107218983949152075, 2136610763952639975, 44868826043005439475, 986129980012277775675

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(4)=21 because only the following three degree-4 permutations do not qualify: (12)(34), (13)(24) and (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

g:=sqrt((1+x)/(1-x))*(product(1+sinh(x^(2*k)/(2*k)),k=1..30)): gser:=series(g, x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=1 or irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i &, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

E.g.f.: sqrt((1+x)/(1-x))*Product_{k>0} (1+sinh(x^(2*k)/(2*k))).

More terms from Emeric Deutsch, Aug 24 2007

A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.

Original entry on oeis.org

1, 2, 16, 416, 20224, 1645312, 196388864, 33279311872, 7427338829824, 2151276556845056, 771086221948223488, 340572557390992900096, 179222835344084459061248, 112158801651454395931426816, 81399358513573250066141937664, 68530340884909785149816189222912

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(2)=16 because there are 8 permutations that do not qualify: (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3) and (143)(2).

Alois P. Heinz, Table of n, a(n) for n = 0..220

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

g:=(product(cosh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x= 0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=0, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 || Mod[j, 2] == 0, multinomial[n, Join[{n - i*j}, Array[i &, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

N=31; x='x+O('x^N);
v0=Vec(serlaplace(1/sqrt(1-x^2)*prod(k=1,N, cosh(x^(2*k-1)/(2*k-1)))));
vector(#v0\2,n,v0[2*n-1]) \\ Joerg Arndt, Jan 03 2011

E.g.f. with interleaved zeros: 1/sqrt(1-x^2)*Product_{k>=1} cosh(x^(2*k-1)/(2*k-1)). - Geoffrey Critzer, Jan 02 2011

More terms from Emeric Deutsch, Aug 24 2007

A130278 Number of degree-n permutations such that number of cycles of size 2k-1 is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 1, 6, 17, 100, 529, 3766, 31121, 276984, 2755553, 29665306, 364627801, 4639937380, 64952094401, 973467571350, 15750475301921, 264870218828656, 4759194994114369, 90124395399063730, 1812001488739061417, 37956199941196210716, 832297726351555617569

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(4)=17 because only the following 7 permutations do not qualify: (1)(2)(3)(4), (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4) and (14)(2)(3).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

g:=(product(1+sinh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x =0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i&, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

E.g.f.: 1/sqrt(1-x^2)*Product_{k>0} (1+sinh(x^(2*k-1)/(2*k-1))).

More terms from Emeric Deutsch, Aug 24 2007

cp's OEIS Frontend

A130126 Number of partitions of n in which each even part has odd multiplicity.

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A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

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A131945 Number of partitions of n where odd parts are distinct or repeated once.

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A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

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A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.

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A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.

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Maple

Mathematica

PARI

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A130278 Number of degree-n permutations such that number of cycles of size 2k-1 is odd (or zero) for every k.

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