cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Original entry on oeis.org

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

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Author

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

Keywords

Comments

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

Examples

			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

References

  • 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).

Links

Crossrefs

Cf. A000041, A001651, A003114, A000726, A109389, A109697, A132462, A132463, A285219, A304047.
Cf. A186099 (log).

Programs

  • Haskell
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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 */
  • PARI
    {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 */

Formula

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

Extensions

More terms from Vladeta Jovovic, Jun 08 2003

A035462 Number of partitions of n into parts 4k-1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 4, 3, 4, 5, 5, 5, 6, 7, 8, 7, 8, 11, 10, 10, 13, 14, 14, 15, 17, 19, 20, 20, 24, 27, 26, 28, 33, 35, 35, 39, 44, 46, 48, 52, 58, 62, 63, 69, 78, 80, 83, 93, 100, 104, 111, 120, 130, 137, 143, 156, 169, 175, 185, 203
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Also, number of partitions into parts 8k+3 or 8k+7.
Also number of partitions of n such that 2k-1 and 2k occur with the same multiplicity. Example: a(18)=3 because we have [8,7,2,1],[6,5,4,3] and [2,2,2,2,2,2,1,1,1,1,1,1]. It is easy to find a bijection between these partitions and those described in the definition. - Emeric Deutsch, Apr 05 2006

Examples

			a(18)=3 because we have [15,3],[11,7] and [3,3,3,3,3,3].

Links

Crossrefs

Cf. A035441-A035468, A050452.
Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), this sequence (m=4), A109700 (m=5), A109702 (m=6), A109708 (m=7).

Programs

  • Maple
    g:=1/product(1-x^(4*i-1),i=1..50): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=1..75); # Emeric Deutsch, Apr 05 2006
  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+3)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 - 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 04 2020 *)

Formula

G.f.: 1/Product_{j>=1} (1 - x^(4*j-1)). - Emeric Deutsch, Apr 05 2006
G.f.: Sum_{n>=0} (x^(3*n) / Product_{k=1..n} (1 - x^(4*k))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k>=n} (1 - x^(4*k+3))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k=0..n} (1 - x^(4*k+3))). - Joerg Arndt, Apr 08 2011
a(n) ~ Pi^(3/4) * exp(Pi*sqrt(n/6)) / (Gamma(1/4) * 2^(13/8) * 3^(3/8) * n^(7/8)) * (1 + (Pi/(96*sqrt(6)) - 21*sqrt(3/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
From Peter Bala, Feb 02 2021: (Start)
G.f.: A(x) = Sum_{n >= 0} x^(n*(4*n-1))/Product_{k = 1..n} ( (1 - x^(4*k))*(1 - x^(4*k-1)) ). (Set z = x^3 and q = x^4 in Mc Laughlin et al., Section 1.3, Entry 7.)
Similarly, A(x) = Sum_{n >= 0} x^(n*(4*n+3))/( (1 - x^3)*Product_{k = 1..n} ((1 - x^(4*k))*(1 - x^(4*k+3))) ). (End)

Extensions

Offset changed by N. J. A. Sloane, Apr 11 2010

A109700 Number of partitions of n into parts each equal to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 3, 4, 2, 2, 3, 5, 4, 3, 3, 6, 6, 6, 4, 6, 7, 9, 7, 7, 8, 11, 11, 11, 9, 12, 14, 16, 13, 14, 16, 21, 20, 19, 18, 24, 26, 27, 24, 27, 31, 36, 34, 34, 35, 43, 45, 47, 43, 49, 55, 62, 58, 59, 63, 75, 77, 77, 75, 87
Offset: 0

Views

Author

Erich Friedman, Aug 07 2005

Keywords

Examples

			a(30)=2 since 30 = 14+4+4+4+4 = 9+9+4+4+4

Links

Crossrefs

Cf. A284103.
Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), this sequence (m=5), A109702 (m=6), A109708 (m=7).

Programs

  • Maple
    g:=1/product(1-x^(4+5*j),j=0..25): gser:=series(g,x=0,95): seq(coeff(gser,x,n),n=0..90); # Emeric Deutsch, Mar 30 2006
  • Mathematica
    nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+4)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

Formula

G.f.: 1/product(1-x^(4+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006
a(n) ~ Gamma(4/5) * exp(Pi*sqrt(2*n/15)) / (2^(19/10) * 3^(2/5) * 5^(1/10) * Pi^(1/5) * n^(9/10)) * (1 - (9*sqrt(6/5)/(5*Pi) + Pi/(120*sqrt(30))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284103(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

Extensions

More terms from Michael Somos, Aug 10 2005

A109708 Number of partitions of n into parts each equal to 6 mod 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 2, 1, 2, 3, 5, 5, 5, 2, 2, 3, 5, 6, 8, 5, 3, 3, 5, 7, 10, 9, 7, 4, 5, 7, 11, 12, 12, 8, 6, 7, 12, 14, 17, 15, 11, 8, 12, 15, 20, 21, 19, 13, 13, 16, 22, 26, 28, 23
Offset: 0

Views

Author

Erich Friedman, Aug 07 2005

Keywords

Examples

			a(45)=3 because we have 45=27+6+6+6=20+13+6+6=13+13+13+6.

Links

Crossrefs

Cf. A284105.
Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), A109702 (m=6), this sequence (m=7).

Programs

  • Maple
    g:=1/product(1-x^(6+7*j),j=0..20): gser:=series(g,x=0,98): seq(coeff(gser,x,n),n=0..95); # Emeric Deutsch, Apr 14 2006
  • Mathematica
    nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+6)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

Formula

G.f.: 1/product(1-x^(6+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(6/7) * exp(Pi*sqrt(2*n/21)) / (2^(27/14) * 3^(3/7) * 7^(1/14) * Pi^(1/7) * n^(13/14)) * (1 - (39*sqrt(3/14)/(7*Pi) + 13*Pi/(168*sqrt(42))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284105(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

Extensions

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015

A284104 a(n) = Sum_{d|n, d == 5 (mod 6)} d.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 11, 0, 0, 0, 5, 0, 17, 0, 0, 5, 0, 11, 23, 0, 5, 0, 0, 0, 29, 5, 0, 0, 11, 17, 40, 0, 0, 0, 0, 5, 41, 0, 0, 11, 5, 23, 47, 0, 0, 5, 17, 0, 53, 0, 16, 0, 0, 29, 59, 5, 0, 0, 0, 0, 70, 11, 0, 17, 23, 40, 71, 0, 0, 0, 5, 0, 88, 0, 0, 5
Offset: 1

Views

Author

Seiichi Manyama, Mar 20 2017

Keywords

Links

Crossrefs

Cf. A086729, A109702.
Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), A284103 (k=5), this sequence (k=6), A284105 (k=7).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 6] == 5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,6]==5&]],{n,80}] (* Harvey P. Dale, Dec 30 2017 *)
  • PARI
    for(n=1, 80, print1(sumdiv(n, d, if(Mod(d,6)==5, d, 0)),", ")) \\ Indranil Ghosh, Mar 21 2017
  • Python
    from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%6==5]) # Indranil Ghosh, Mar 21 2017

Formula

G.f.: Sum_{k>=1} (6*k - 1)*x^(6*k-1)/(1 - x^(6*k-1)). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/72 = 0.137077... (A086729). - Amiram Eldar, Nov 26 2023

A339089 Number of compositions (ordered partitions) of n into distinct parts congruent to 5 mod 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 4, 1, 0, 0, 0, 6, 4, 1, 0, 0, 0, 6, 6, 1, 0, 0, 0, 12, 6, 1, 0, 0, 0, 18, 8, 1, 0, 0, 24, 24, 8, 1, 0, 0, 24, 30, 10, 1, 0, 0, 48, 42, 10, 1, 0, 0, 72, 48, 12, 1, 0, 0, 120, 60, 12, 1, 0, 120, 144
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(33) = 6 because we have [17, 11, 5], [17, 5, 11], [11, 17, 5], [11, 5, 17], [5, 17, 11] and [5, 11, 17].

Links

Crossrefs

Cf. A016969, A017837, A032020, A032021, A109702, A281244, A337547, A337548, A339059, A339060, A339086, A339087, A339088.

Programs

  • Mathematica
    nmax = 86; CoefficientList[Series[Sum[k! x^(k (3 k + 2))/Product[1 - x^(6 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)).

A304885 Expansion of Product_{k>=1} 1/(1-x^(3*k-2)) * Product_{k>=1} 1/(1-x^(6*k-1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 8, 10, 12, 14, 17, 21, 25, 30, 35, 41, 49, 58, 68, 79, 92, 107, 124, 144, 166, 191, 220, 252, 289, 331, 378, 431, 490, 557, 632, 717, 812, 917, 1035, 1167, 1315, 1480, 1663, 1866, 2092, 2344, 2624, 2934, 3277, 3656, 4076, 4542, 5056
Offset: 0

Views

Author

Seiichi Manyama, May 20 2018

Keywords

Links

Crossrefs

Cf. A035382, A109702, A304883.

Programs

  • Maple
    seq(coeff(series(mul(1/(1-x^(3*k-2)),k=1..n)*mul(1/(1-x^(6*k-1)),k=1..n), x,70),x,n),n=0..60); # Muniru A Asiru, May 21 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/((1-x^(3*k-2)) * (1-x^(6*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 21 2018 *)

Formula

G.f.: Sum_{j>=0} x^(j*(3*j-1)/2)*(Product_{k=1..j} (1-x^(6*k-4)))/(Product_{k=1..3*j} (1-x^k)).
a(n) ~ exp(Pi*sqrt(n/3)) * Pi^(2/3) / (2 * 3^(2/3) * Gamma(1/3) * n^(5/6)). - Vaclav Kotesovec, May 21 2018
Showing 1-7 of 7 results.

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