A096938 -id:A096938 - cp's OEIS frontend

A003105 Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.

1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23, 26, 30, 34, 38, 42, 47, 53, 60, 67, 74, 82, 91, 102, 114, 126, 139, 153, 169, 187, 207, 228, 250, 274, 301, 331, 364, 399, 436, 476, 520, 569, 622, 679, 739, 804, 875, 953, 1038, 1128, 1224, 1327

Offset: 0

N. J. A. Sloane, Herman P. Robinson

There are many (at least 8) equivalent definitions of this sequence (besides the comments below, see also Schur, Alladi, Andrews). - N. J. A. Sloane, Jun 17 2011
Coefficients of replicable function number 72e. - N. J. A. Sloane, Jun 10 2015
Also number of partitions of n into odd parts in which no part appears more than twice, cf. A070048 and A096938. - Vladeta Jovovic, Jan 18 2005
Also number of partitions of n into distinct parts congruent to 1 or 2 modulo 3. (Follows from second g.f.) - N. Sato, Jul 20 2005
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A262928 and A261612. - Vaclav Kotesovec, Jan 13 2017
Convolution of A109702 and A109701. - Vaclav Kotesovec, Jan 21 2017

			G.f: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
T72e = 1/q + q^11 + q^23 + q^35 + q^47 + 2*q^59 + 2*q^71 + 3*q^83 + ...
The logarithm of the g.f. begins:
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 + x^9/9 + 6*x^10/10 + 12*x^11/11 + x^12/12 + ... + A186099(n)*x^n/n + ... . - _Paul D. Hanna_, Feb 17 2013

K. Alladi, Refinements of Rogers-Ramanujan type identities. In Special Functions, q-Series and Related Topics (Toronto, ON, 1995), 1-35, Fields Inst. Commun., 14, Amer. Math. Soc., Providence, RI, 1997.
G. E. Andrews, Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. In q-Series From a Contemporary Perspective (South Hadley, MA, 1998), 45-56, Contemp. Math., 254, Amer. Math. Soc., Providence, RI, 2000.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
I. Schur, Zur Additiven Zahlentheorie, Ges. Abh., Vol. 2, Springer, pp. 43-50.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 201 terms from R. Zumkeller)
K. Alladi and B. Gordon, Generalizations of Schur's partition theorem, Manuscr. Math. 79 (1993) 113-126.
K. Alladi and B. Gordon, Schur's partition theorem, companions, refinements and generalizations, Trans. Amer. Math. Soc. 347 (1995) 1591-1608.
G. E. Andrews, K. Alladi, B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, J. Reine Angew. Math. 460 (1995) 165-188.
D. M. Bressoud, A combinatorial proof of Schur's 1926 partition theorem, Proc. Amer, Math. Soc. 79 (1980) 338-340.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. Vol. 225 (1967), 154-190.
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, p. 12.
Padmavathamma, R. Raghavendra and B. M. Chandrashekara, A new bijective proof of a partition theorem of K. Alladi, Discrete Math., 237 (2004), 125-128.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Schur's Partition Theorem
James J. Y. Zhao, A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem, Elect. J. Combin, Volume 22, Issue 1 (2015) Paper #P1.68.
Index entries for McKay-Thompson series for Monster simple group

Cf. A000041, A001651, A003114, A000726, A109389, A109697, A132462, A132463, A285219, A304047.

Cf. A186099 (log).

a003105 n = p 1 n where
   p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+2) 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

with(combinat);
A:=proc(n) local i, j, t3, t2, t1;
    t2:=0;
    t1:=firstpart(n);
    for j from 1 to numbpart(n)+2 do
        t3:=1;
        for i from 1 to nops(t1) do
            if (t1[i] mod 6) <> 1 and (t1[i] mod 6) <> 5 then t3:=0; fi;
        od;
        if t3=1 then t2:=t2+1; fi;
        if nops(t1) = 1 then RETURN(t2); fi;
        t1:=nextpart(t1);
    od;
end;
# brute-force Maple program from N. J. A. Sloane, Jun 17 2011

max = 63; f[x_] := 1/Product[1 - x^k + x^(2k), {k, 0, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 01 2011, after Vladeta Jovovic *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] / QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Jul 05 2014 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 3] != 0, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 63; kmax = nmax/6;
s = Flatten[{Range[0, kmax]*6 + 1}~Join~{Range[kmax]*6 - 1}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 09 2005 */

{S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}
{a(n)=polcoeff(exp(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n)),n)}
for(n=0,60,print1(a(n),", "))
/* Paul D. Hanna, Feb 17 2013 */

G.f.: 1/Product_{k>=0} (1-x^(6*k+1))*(1-x^(6*k+5)) = Product_{k>=0} (1+x^(3*k+1))*(1+x^(3*k+2)) = 1/Product_{k>=0} (1-x^k+x^(2*k)). - Vladeta Jovovic, Jun 08 2003
Expansion of chi(-x^3) / chi(-x) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Mar 04 2012
Expansion of f(x, x^2) / f(-x^3) = f(-x^6) / f(-x, -x^5) in powers of x where f() is Ramanujan theta function. - Michael Somos, Jul 05 2014
Expansion of q^(1/12) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jan 09 2005
Euler transform of period 6 sequence [1, 0, 0, 0, 1, 0, ...]. - Michael Somos, Jan 09 2005
Given g.f. A(x), then B(q) = (A(q^12) / q)^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v^4 + (1 - u^3) * v^3 + 6*u^2*v^2 + (u^4 - u)*v + u^3. - Michael Somos, Jan 09 2005
The logarithmic derivative equals A186099. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} A186099(n) * x^n/n ) where A186099(n) = sum of divisors of n congruent to 1 or 5 mod 6. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} S(n,x) * x^n/n ) where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013
a(n) ~ Pi*sqrt(2) / sqrt(3*(12*n-1)) * BesselI(1, Pi*sqrt(12*n-1) / (3*sqrt(6))) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)) * (1 - (9/(8*Pi) + Pi/36)/sqrt(2*n) + (5 - 135/(4*Pi^2) + Pi^2/81)/(64*n)). - Vaclav Kotesovec, Aug 23 2015, extended Jan 09 2017
a(n) = (1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

More terms from Vladeta Jovovic, Jun 08 2003

A035959 Number of partitions of n in which no parts are multiples of 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, 127, 164, 205, 262, 325, 409, 505, 628, 769, 950, 1156, 1414, 1713, 2081, 2505, 3026, 3625, 4352, 5192, 6200, 7364, 8756, 10357, 12258, 14450, 17034, 20006, 23500, 27510, 32200, 37582, 43846, 51022

Offset: 0

Olivier Gérard

Also number of partitions with at most 4 parts of size 1 and differences between parts at distance 6 are greater than 1.
Also number of partitions of n where no part appears more than four times.
Case k=7, i=5 of Gordon Theorem.

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 13*x^7 + 19*x^8 + ...
G.f. = q + q^7 + 2*q^13 + 3*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 13*q^43 + ...
a(6) counts these partitions: 6, 42, 411, 33, 321, 3111, 2211, 21111, 111111. - _Clark Kimberling_, Mar 09 2014

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
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, p. 15.
G. N. Watson, Ramanujans Vermutung ueber Zerfaellungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression Y = C/B in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]
Eric Weisstein's World of Mathematics, Partition Function b_k.
Wikipedia, Glaisher's Theorem

Cf. A061198, A061199, A047201.
Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10).
Cf. A096938, A133563, A320607.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a035959 = p a047201_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 17 2011

max = 47; f[x_] := (x^5-1)/(x-1); g[x_] := Product[f[x^k], {k, 1, max}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 29 2011, after Michael Somos *)
t = Flatten[Table[5 n + r, {n, 0, 60}, {r, 1, 4}]]; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 8}] (* shows partitions *)
a[n_] := Length@p@n; a /@ Range[0, 50] (* Clark Kimberling, Mar 09 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^5]/QP[q] + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 5], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)
Table[Count[IntegerPartitions[n],?(NoneTrue[Mod[#,5]==0&])],{n,0,50}] (* _Harvey P. Dale, Dec 25 2021 *)

{a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) / eta(x + x * O(x^n)), n))}; /* Michael Somos, May 28 2006 */

G.f.: Product_{j>=1} (1 + x^j + x^2j + x^3j + x^4j). - Jon Perry, Mar 30 2004
G.f.: Product_{n>0, n==1, 2, 3, 4 mod 5} 1/(1-q^n).
Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - u*v - 5*u^2*v^2. - Michael Somos, May 28 2006
Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v + 5*v^2*(u + w) - (u^2 + u*w + w^2). - Michael Somos, May 28 2006
Euler transform of period 5 sequence [ 1, 1, 1, 1, 0, ...]. - Michael Somos, May 28 2006
G.f.: Product_{k > 0} P5(x^k) where P5 is 5th cyclotomic polynomial.
Convolution inverse is A145466. - Michael Somos, Jun 26 2014
a(n) ~ 2*Pi * BesselI(1, 2*sqrt((6*n + 1)/5) * Pi/3) / (5*sqrt(6*n + 1)) ~ exp(2*Pi*sqrt(2*n/15)) / (3^(1/4) * 10^(3/4) * n^(3/4)) * (1 + (Pi/(3*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(2*n) + (Pi^2/540 - 225/(1024*Pi^2) - 5/32) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A116073(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k))/(k*(1 - x^(5*k)))). - Ilya Gutkovskiy, Aug 15 2018

A070048 Number of partitions of n into odd parts in which no part appears more than thrice.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 18, 21, 24, 27, 32, 36, 41, 48, 54, 61, 70, 78, 88, 100, 112, 127, 143, 159, 179, 199, 222, 248, 276, 308, 342, 380, 421, 465, 516, 570, 629, 697, 767, 845, 932, 1022, 1124, 1236, 1355, 1488, 1631, 1785, 1954, 2136

Offset: 0

N. J. A. Sloane, May 09 2002

nonn

Also number of partitions of n into distinct parts in which no part is multiple of 4. - Vladeta Jovovic, Jul 31 2004
McKay-Thompson series of class 64a for the Monster group.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
T64a = 1/q + q^7 + q^15 + 2*q^23 + q^31 + 2*q^39 + 3*q^47 + 3*q^55 + 4*q^63 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..220 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.
Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
M. D. Hirschhorn, J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
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, p. 12.
Index entries for McKay-Thompson series for Monster simple group

Cf. A042968, A001935, A261734.

Cf. A000700 (m=2), A003105 (m=3), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).

a070048 = p a042968_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Oct 01 2012

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^4], {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] / (QPochhammer[ x] QPochhammer[ x^8]), {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)

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

G.f.: Product_{i>0} (1+x^i)/(1+x^(4*i)). - Vladeta Jovovic, Jul 31 2004
Expansion of chi(x) * chi(x^2) = psi(x) / psi(-x^2) = phi(-x^4) / psi(-x) = chi(-x^4) / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jul 01 2014
Expansion of q^(1/8) * eta(q^2) * eta(q^4) / (eta(q) * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [1, 0, 1, -1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) + 3*u*v. - Michael Somos, Jul 01 2014
G.f.: Product_{k>0} (1 - x^(8*k - 4)) / (1 - x^(2*k - 1)).
a(n) ~ exp(sqrt(n)*Pi/2) / (4*n^(3/4)) * (1 - (3/(4*Pi) + Pi/32) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

Additional comments from Michael Somos, Dec 04 2002

A097793 McKay-Thompson series of class 56B for the Monster group.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 10, 12, 14, 17, 21, 24, 28, 34, 39, 46, 53, 61, 71, 82, 94, 108, 124, 142, 162, 185, 210, 238, 271, 306, 345, 390, 439, 494, 556, 623, 698, 783, 875, 977, 1092, 1216, 1354, 1508, 1674, 1859, 2064, 2286, 2532, 2803, 3098, 3424

Offset: 0

Michael Somos, Aug 24 2004

nonn

Number of partitions of n into distinct parts not divisible by 7.

Also McKay-Thompson series of class 56C for Monster. - Michel Marcus, Feb 19 2014

			1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 +...
T56B = 1/q + q^3 + q^7 + 2q^11 + 2q^15 + 3q^19 + 4q^23 + 4q^27 +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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, p. 12.
Index entries for McKay-Thompson series for Monster simple group

Cf. A113297.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A261771 (m=8), A112193 (m=9), A261772 (m=10).

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^7]/(QP[q]*QP[q^14])) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, 1 + x^k, 1 + A) / prod( k=1, n\7, 1 + x^(7*k), 1 + A), n))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^14 + A)), n))}

Euler transform of period 14 sequence [ 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, ...].
Expansion of q^(1/4) * eta(q^2) * eta(q^7) / (eta(q) * eta(q^14)) in powers of q.
G.f.: Product_{k>0} (1 + x^k) / (1 + x^(7*k)).
a(n) ~ exp(Pi*sqrt(2*n/7)) / (2 * 14^(1/4) * n^(3/4)) * (1 - (3*sqrt(7)/ (8*Pi*sqrt(2)) + Pi/(4*sqrt(14))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

A112193 Coefficients of replicable function number "54b".

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 95, 110, 127, 145, 167, 191, 218, 249, 283, 322, 365, 414, 469, 529, 597, 673, 757, 851, 955, 1071, 1199, 1341, 1499, 1673, 1865, 2078, 2313, 2572, 2857, 3171, 3517, 3897

Offset: 0

Michael Somos, Aug 28 2005

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 9. - Joerg Arndt, Aug 31 2015

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)/(1 + x^(m*k)), then a(n) ~ exp(Pi*sqrt((m-1)*n/(3*m))) * (m-1)^(1/4) / (2^(3/2) * 3^(1/4) * m^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + ... _Michael Somos_, Oct 06 2019
G.f. = q^-1 + q^2 + q^5 + 2*q^8 + 2*q^11 + 3*q^14 + 4*q^17 + 5*q^20 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
Index entries for McKay-Thompson series for Monster simple group

Cf. A261733.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A261772 (m=10).

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 9*j*(j+1))(iquo(i, 9, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(9*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
b[n_, i_] := b[n, i] = Module[{q, r}, {q, r} = QuotientRemainder[i, 9]; If[2*n > i*(i+1) - 9*q*(q+1), 0, If[n == 0, 1, b[n, i-1] + If[i>n || r == 0, 0, b[n-i, i-1]]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 07 2016, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]), {q, 0, n}]; (* Michael Somos, Oct 06 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)), n))}; /* Michael Somos, Oct 06 2019 */

a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (6^(3/4) * n^(3/4)) * (1 - (9*sqrt(3)/ (16*Pi*sqrt(2)) + sqrt(2)*Pi/(9*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017
From Michael Somos, Oct 06 2019: (Start)
Expansion of q^(1/3) * eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...].
G.f. is a period 1 Fourier Series which satisifies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + u*v) * (u^3 + v^3) - u*v * (1 + u^2*v^2).
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (w^2 - v) * (u^2 - v) - 2*u*v*w.
Convolution inverse of A261733.
(End)

A261770 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(6*k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 11, 13, 16, 19, 22, 26, 30, 35, 41, 47, 55, 63, 73, 84, 96, 110, 125, 143, 162, 184, 208, 235, 266, 300, 338, 380, 427, 479, 536, 600, 670, 748, 834, 929, 1034, 1149, 1277, 1417, 1571, 1740, 1925, 2129, 2351, 2596, 2863

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 6. - Joerg Arndt, Aug 31 2015

Alois P. Heinz, 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, p. 12.

Cf. A261736.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).
Column k=6 of A290307.

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 6*j*(j+1))(iquo(i, 6, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(5*n/2)/3) * 5^(1/4) / (2^(7/4) * sqrt(3) * n^(3/4)) * (1 - (9/(4*Pi*sqrt(10)) + 5*Pi*sqrt(5/2)/144) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(12*k-6))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

A261771 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(8*k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 13, 15, 18, 22, 26, 30, 36, 42, 49, 58, 67, 77, 89, 103, 118, 136, 156, 177, 203, 231, 263, 299, 338, 383, 433, 489, 550, 620, 696, 781, 877, 981, 1097, 1227, 1369, 1526, 1702, 1893, 2104, 2339, 2595, 2877, 3189, 3530

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 8.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A261735.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A112193 (m=9), A261772 (m=10).
Column k=8 of A290307.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
       [0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1]
       [1+irem(d, 16)], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

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

a(n) ~ exp(Pi*sqrt(7*n/6)/2) * 7^(1/4) / (4 * 6^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/ (2*Pi*sqrt(14)) + 7*Pi*sqrt(7)/(96*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(16*k-8))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

A261772 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(10*k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 20, 24, 28, 33, 40, 46, 54, 64, 74, 86, 100, 115, 133, 154, 176, 202, 231, 263, 300, 342, 388, 440, 499, 563, 636, 718, 808, 909, 1022, 1146, 1284, 1439, 1608, 1797, 2006, 2236, 2490, 2772, 3081, 3422, 3800, 4212

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 10.

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)/(1 + x^(m*k)), then a(n) ~ exp(Pi*sqrt((m-1)*n/(3*m))) * (m-1)^(1/4) / (2^(3/2) * 3^(1/4) * m^(1/4) * n^(3/4)) * (1 - (3*sqrt(3*m)/(8*Pi*sqrt(m-1)) + (m-1)^(3/2)*Pi/(48*sqrt(3*m))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A145707.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9).
Column k=10 of A290307.

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 10*j*(j+1))(iquo(i, 10, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(3*n/10)) * 3^(1/4) / (2^(7/4) * 5^(1/4) * n^(3/4)) * (1 - (sqrt(15)/(4*Pi*sqrt(2)) + 3*Pi*sqrt(3)/(16*sqrt(10))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(20*k-10))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 12, 15, 21, 25, 30, 39, 46, 56, 72, 85, 101, 125, 147, 175, 215, 252, 296, 356, 415, 487, 582, 676, 786, 927, 1072, 1244, 1460, 1682, 1939, 2255, 2588, 2976, 3446, 3942, 4510, 5189, 5916, 6751, 7739, 8797, 9999, 11406, 12927, 14657

Offset: 0

Noureddine Chair, Aug 19 2004

nonn

Also, number of partitions of n in which the distinct parts are prime to 3 and the unrestricted parts are multiples of 3.

The inverted graded parafermionic partition function. This g.f. is a generalization of A003105, A006950 and A096938

			a(11) = 15 because we can write 11 = 10+1 = 8+2+1 = 7+4 = 5+4+2 (parts do not contain multiple of 3) = 9+2 = 8+3 = 7+3+1 = 6+5 = 6+4+1 = 6+3+2 = 5+3+3 = 5+3+2+1 = 4+3+3+1 = 3+3+3+2.
1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + ...
q^-5 + q^19 + q^43 + 2*q^67 + 2*q^91 + 3*q^115 + 5*q^139 + 6*q^163 + 7*q^187 + ...

T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 23-June-2013.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.

Cf. A047273, A056970, A097451, A098884.

a096981 = p $ tail a047273_list where
   p _  0         = 1
   p ks'@(k:ks) m = if k > m then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013

series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)), k=1..150), x=0,100);

CoefficientList[ Series[ Product[ 1/(1 - x^k + x^(2k) - x^(3k) + x^(4k) - x^(5k)), {k, 55}], {x, 0, 53}], x] (* Robert G. Wilson v, Aug 21 2004 *)
nmax = 100; CoefficientList[Series[x^3*QPochhammer[-1/x^2, x^3] * QPochhammer[-1/x, x^3]/((1 + x)*(1 + x^2) * QPochhammer[x^3, x^3]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / (eta(x + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 08 2012 */

Expansion of q^(5/24) * eta(q^2) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012
Euler transform of period 6 sequence [1, 0, 1, 0, 1, 1, ...]. - Vladeta Jovovic, Aug 20 2004
G.f.: 1/product_{k>=1}(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)) = Product_{k>=1}(1+x^(3*k-1))(1+x^(3*k-2))/(1-x^(3*k)).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(6)*n). - Vaclav Kotesovec, Aug 31 2015

Better definition from Vladeta Jovovic, Aug 20 2004
More terms from Robert G. Wilson v, Aug 21 2004
Incorrect b-file replaced by Vaclav Kotesovec, Aug 31 2015

A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 3, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 4, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 5, 3, 0

Offset: 0

Ilya Gutkovskiy, Jul 26 2017

A(n,k) is the number of partitions of n into distinct parts where no part is a multiple of k.

			Square array begins:
  1,  1,  1,  1,  1,  1, ...
  0,  1,  1,  1,  1,  1, ...
  0,  0,  1,  1,  1,  1, ...
  0,  1,  1,  2,  2,  2, ...
  0,  1,  1,  1,  2,  2, ...
  0,  1,  2,  2,  2,  3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=1-10 give: A000007, A000700, A003105, A070048, A096938, A261770, A097793, A261771, A112193, A261772.

Cf. A286653.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)/(1 + x^(i k)), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x]/QPochhammer[-1, x^k], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten

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

For asymptotics of column k see comment from Vaclav Kotesovec in A261772.

cp's OEIS Frontend

A003105 Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.

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A035959 Number of partitions of n in which no parts are multiples of 5.

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A070048 Number of partitions of n into odd parts in which no part appears more than thrice.

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A097793 McKay-Thompson series of class 56B for the Monster group.

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A112193 Coefficients of replicable function number "54b".

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A261770 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(6*k)).

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