A280937 -id:A280937 - cp's OEIS frontend

A070047 Number of partitions of n in which no part appears more than twice and no two parts differ by 1.

1, 1, 2, 1, 3, 3, 5, 5, 8, 8, 12, 12, 19, 19, 27, 28, 39, 41, 55, 58, 77, 82, 106, 113, 145, 156, 196, 210, 262, 283, 348, 376, 459, 497, 600, 651, 781, 849, 1009, 1097, 1298, 1413, 1660, 1807, 2113, 2302, 2676, 2916, 3377, 3681, 4242, 4623, 5309, 5787, 6619

Offset: 0

N. J. A. Sloane, May 09 2002

nonn

Coefficients in expansion of permanent of infinite tridiagonal matrix: matrix([[1, x, 0, 0, 0, ...], [1+x, 1, x^2, 0, 0, ...], [0, 1+x^2, 1, x^3, 0, ...], [0, 0, 1+x^3, 1, x^4, ...], ...]). - Vladeta Jovovic, Jul 18 2004
Number of partitions of n into non-multiples of 3 in which no two parts differ by 1 (see the Andrews-Lewis reference). Example: a(6)=5 because we have 51,42,411,222,111111. - Emeric Deutsch, May 19 2008
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + 2*x^2 + x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 5*x^7 + 8*x^8 + 8*x^9 + 12*x^10 + ...
G.f. = 1/q + q^23 + 2*q^47 + q^71 + 3*q^95 + 3*q^119 + 5*q^143 + 5*q^167 + 8*q^191 + ...
a(6)=5 because we have 6,51,42,411,33.

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..120 from Reinhard Zumkeller)
G. E. Andrews and R. P. Lewis, An algebraic identity of F. H. Jackson and its implications for partitions, Discrete Math., 232 (2001), 77-83. see equations (3.1) and (3.2)
Bin Lan and James A. Sellers, Properties of a Restricted Binary Partition Function a la Andrews and Lewis, Electronic Journal of Combinatorial Number Theory, Volume 15 #A23.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Wikipedia, Bailey pair
Mingjia Yang and Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

Cf. A233006, A000700, A108961, A108962, A271661, A280937, A280938.

a070047 n = p 1 n where
   p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+1) m
   q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m
-- Reinhard Zumkeller, Nov 12 2011

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-2), j=1..min(n/i, 2))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 18 2013

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Product[ (1 - x^(6 k - 3))^2 (1 - x^(6 k)), {k, Ceiling[ n/6]}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]]; (* Michael Somos, Jun 02 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Dec 03 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-2], {j, 1, Min[n/i, 2]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 04 2015, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[Product[1 / ( (1-x^(3*k-2)) * (1-x^(3*k-1)) * (1 + x^(3*k)) ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jun 02 2011 */

Expansion of phi(-x^3) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions. - Michael Somos, Jun 02 2011
Expansion of q^(1/24) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Dec 04 2002
Euler transform of period 6 sequence [ 1, 1, -1, 1, 1, 0, ...]. - Michael Somos, Dec 04 2002
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = (2/3)^(1/2) g(t) where q = exp(2 Pi i t) and g is the g.f. of A233006.
G.f.: Prod_{k>0} (1 - x^(6*k - 3))^2 * (1 - x^(6*k)) / (1 - x^k).
G.f.: Prod_{n>0}[(1-q^(6n-3))/[(1-q^(3n-2))(1-q^(3n-1))]]. - Emeric Deutsch, May 19 2008
a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)/2)) / sqrt(3*(24*n-1)) ~ exp(Pi*sqrt(n/3)) / (2*3^(3/4)*n^(3/4)) * (1 - (3*sqrt(3)/(8*Pi) + Pi/(48*sqrt(3)))/sqrt(n) + (Pi^2/13824 - 45/(128*Pi^2) + 5/128)/n). - Vaclav Kotesovec, Sep 02 2015, extended Jan 11 2017
G.f.: Product_{n >= 1} (1 + q^n + q^(2*n))/(1 + q^(3*n)). - Peter Bala, Jan 04 2025

Additional comments from Michael Somos, Dec 04 2002

A108961 Number of partitions that are "2-close" to being self-conjugate.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 7, 9, 12, 16, 20, 26, 33, 41, 51, 64, 79, 97, 119, 144, 175, 212, 254, 305, 365, 434, 516, 612, 722, 851, 1002, 1174, 1375, 1607, 1872, 2179, 2531, 2933, 3395, 3923, 4524, 5211, 5994, 6881, 7891, 9038, 10334, 11804, 13467, 15341, 17460, 19849

Offset: 0

John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005

nonn

Let (a1,a2,a3,...ad:b1,b2,b3,...bd) be the Frobenius symbol for a partition pi of n. Then pi is m-close to being self-conjugate if for all k, |ak-bk| <= m.

Convolution of A070048 and A035457. - Vaclav Kotesovec, Nov 13 2016

			1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 12*x^8 + 16*x^9 + 20*x^10 + ...
1/q + q^23 + 2*q^47 + 3*q^71 + 3*q^95 + 5*q^119 + 7*q^143 + 9*q^167 + 12*q^191 + ...

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Wikipedia, Bailey pair

Cf. A000700 for m=0 (self-conjugate), A070047 for m=1, A108962 for m=3, A271661 for m=4, A280937 for m=5, A280938 for m=6.

nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(2*k)) / (1 + x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A) * eta(x^8 + A)), n))} /* Michael Somos, Oct 17 2006 */

Define the Dedekind eta function = q^1/24. Product(1-q^k), k >=1. Then the number of m-close partitions is q^(1/24).(m+2)^2/(1.(2m+4)) (where m denotes eta(q^m)).
Expansion of q^(1/24) * eta(q^4)^2 / (eta(q) * eta(q^8)) in powers of q. - Michael Somos, Oct 17 2006
Expansion of chi(x^2) * chi(x) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function (see A000700). - Michael Somos, Oct 17 2006 [corrected by Peter Bala, Oct 09 2023]
Euler transform of period 8 sequence [ 1, 1, 1, -1, 1, 1, 1, 0, ...]. - Michael Somos, Oct 17 2006
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k)) / (1 + x^(4*k)). - Michael Somos, Oct 17 2006
a(n) ~ Pi * BesselI(1, Pi * sqrt(5*(24*n-1)/2)/12) / (2*sqrt((24*n-1)/5)) ~ 5^(1/4) * exp(sqrt(5*n/3)*Pi/2) / (2^(5/2) * 3^(1/4) * n^(3/4)) * (1 -(3*sqrt(3) / (4*Pi*sqrt(5)) + Pi*sqrt(5)/(96*sqrt(3)))/sqrt(n) + (5*Pi^2/55296 - 9/(32*Pi^2) + 5/128)/n). - Vaclav Kotesovec, Nov 13 2016, extended Jan 11 2017

A108962 Number of partitions that are "3-close" to being self-conjugate.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 9, 11, 16, 20, 28, 34, 47, 57, 75, 92, 119, 143, 183, 220, 277, 332, 412, 491, 605, 718, 874, 1036, 1252, 1475, 1772, 2082, 2483, 2909, 3450, 4027, 4755, 5533, 6499, 7545, 8826, 10213, 11904, 13741, 15955, 18372, 21262, 24422

Offset: 0

John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005

nonn

Let (a1,a2,a3,...ad:b1,b2,b3,...bd) be the Frobenius symbol for a partition pi of n. Then pi is m-close to being self-conjugate if for all k, |ak-bk| <= m.

			1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 11*x^7 + 16*x^8 + 20*x^9 + 28*x^10 + ...
1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 5*q^119 + 9*q^143 + 11*q^167 + 16*q^191 + ...

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Wikipedia, Bailey pair

Cf. A000700 for m=0 (self-conjugate), A070047 for m=1, A108961 for m=2, A271661 for m=4, A280937 for m=5, A280938 for m=6.

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^2 / ((1 - x^k) * (1 - x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^2 / (eta(x + A) * eta(x^10 + A)), n))} /* Michael Somos, Jun 08 2012 */

Define the Dedekind eta function = q^1/24. Product(1-q^k), k >=1. Then the number of m-close partitions is q^(1/24).(m+2)^2/(1.(2m+4)) (where m denotes eta(q^m)).
From Vaclav Kotesovec, Nov 13 2016, extended Jan 11 2017: (Start)
a(n, m) ~ exp(Pi*sqrt((2*m+1)*n/(3*(m+2)))) * (2*m+1)^(1/4) / (2*3^(1/4)*(m+2)^(3/4)*n^(3/4)).
For m=3, a(n) ~ 7^(1/4) * exp(sqrt(7*n/15)*Pi) / (2*3^(1/4)*5^(3/4)*n^(3/4)) * (1 -(3*sqrt(15)/(8*Pi*sqrt(7)) + Pi*sqrt(7)/(48*sqrt(15)))/sqrt(n) + (7*Pi^2/69120 - 225/(896*Pi^2) + 5/128)/n).
(End)
a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)*7/10)) / (5*sqrt((24*n-1)/7)). - Vaclav Kotesovec, Jan 11 2017

A271661 Expansion of phi(-x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 13, 18, 24, 32, 42, 55, 71, 91, 116, 147, 185, 231, 288, 357, 440, 540, 661, 807, 980, 1186, 1432, 1724, 2069, 2476, 2956, 3521, 4183, 4958, 5865, 6923, 8155, 9587, 11251, 13180, 15411, 17990, 20967, 24399, 28348, 32886, 38098, 44075

Offset: 0

Michael Somos, Apr 11 2016

nonn

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

Partitions which are "4-close" to being self-conjugate; see A108961 for the definition. - Arvind Ayyer, Apr 13 2021

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 18*x^8 + ...
G.f. = q^-1 + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 9*q^143 + 13*q^167 + ..

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988. See page 6 equation 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
D. M. Bressoud, Extension of the partition sieve, J. Number Theory 12 no. 1 (1980), 87-100.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Wikipedia, Bailey pair

Cf. A000700, A070047, A108961, A108962, A280937, A280938.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^6] / QPochhammer[ x], {x, 0, n}];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x, x, 2 k] // FunctionExpand, {k, 0, Sqrt@n}], {x, 0, n}]];
nmax = 50; CoefficientList[Series[Product[(1-x^(6*k)) / ((1-x^k) * (1+x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 18 2016 *)

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

Expansion of f(x^2, x^4) / psi(-x) in powers of x where psi(), f(, ) are Ramanujan theta functions.
Expansion of q^(1/24) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k^2) (-x, x^2)k / (x){2*k}.
a(n) ~ Pi * BesselI(1, Pi*sqrt(24*n-1)/(4*sqrt(3))) / sqrt(24*n-1) ~ exp(sqrt(n/2)*Pi) / (2^(7/4)*sqrt(3)*n^(3/4)) * (1 - (3/(4*Pi) + Pi/48)/sqrt(2*n) + (5/128 - 15/(64*Pi^2) + Pi^2/9216)/n). - Vaclav Kotesovec, Apr 18 2016, extended Jan 10 2017

A280938 Expansion of Product_{k>=1} (1 - x^(8(2k-1))) * (1 - x^(8*k)) / (1 - x^k).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 20, 28, 38, 50, 67, 87, 113, 146, 187, 237, 301, 378, 473, 590, 732, 903, 1113, 1364, 1666, 2030, 2464, 2981, 3600, 4332, 5201, 6229, 7442, 8869, 10551, 12521, 14829, 17531, 20684, 24357, 28638, 33607, 39375, 46062, 53798, 62736

Offset: 0

Vaclav Kotesovec, Jan 11 2017

nonn

In general, if r>=2 and g.f. = Product_{k>=1} (1-x^(r*(2*k-1))) * (1-x^(r*k)) / (1-x^k), then
a(n, r) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)*(2*r-3)/(2*r))) / (r*sqrt((24*n-1)/(2*r-3))).
a(n, r) ~ exp(Pi * sqrt((2/3 - 1/r)*n)) * (2*r-3)^(1/4) / (2 * 3^(1/4) * r^(3/4) * n^(3/4)) * (1 -(3*sqrt(3*r)/(8*Pi*sqrt(2*r-3)) + Pi*sqrt(2*r-3)/(48*sqrt(3*r))) / sqrt(n) + (Pi^2*(2*r-3)/(13824*r) - 45*r/(128*Pi^2*(2*r-3)) + 5/128)/n).

D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Wikipedia, Bailey pair.

Cf. A000700 (r=2), A070047 (r=3), A108961 (r=4), A108962 (r=5), A271661 (r=6), A280937 (r=7).

nmax = 50; CoefficientList[Series[Product[(1-x^(8*(2*k-1))) * (1-x^(8*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ Pi * BesselI(1, Pi * sqrt(13*(24*n-1))/24) / (4*sqrt((24*n-1)/13)).

a(n) ~ exp(Pi*sqrt(13*n/6)/2) * 13^(1/4) / (2^(13/4) * 3^(1/4) * n^(3/4)) * (1 -(3*sqrt(3)/(2*Pi*sqrt(26)) + Pi*sqrt(13)/(96*sqrt(6)))/sqrt(n) + (13*Pi^2/110592 - 45/(208*Pi^2) + 5/128)/n).

cp's OEIS Frontend

A070047 Number of partitions of n in which no part appears more than twice and no two parts differ by 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A108961 Number of partitions that are "2-close" to being self-conjugate.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A108962 Number of partitions that are "3-close" to being self-conjugate.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A271661 Expansion of phi(-x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A280938 Expansion of Product_{k>=1} (1 - x^(8*(2*k-1))) * (1 - x^(8*k)) / (1 - x^k).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A280938 Expansion of Product_{k>=1} (1 - x^(8(2k-1))) * (1 - x^(8*k)) / (1 - x^k).