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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.

A003106 Number of partitions of n into parts 5k+2 or 5k+3.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 16, 20, 22, 26, 29, 35, 38, 45, 50, 58, 64, 75, 82, 95, 105, 120, 133, 152, 167, 190, 210, 237, 261, 295, 324, 364, 401, 448, 493, 551, 604, 673, 739, 820, 899, 997, 1091, 1207, 1321, 1457, 1593, 1756, 1916, 2108, 2301
Offset: 0

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Author

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

Keywords

Comments

Expansion of Rogers-Ramanujan function H(x) in powers of x.
Also number of partitions of n such that the number of parts is greater by one than the smallest part. - Vladeta Jovovic, Mar 04 2006
Example: a(10)=4 because we have [9, 1], [6, 2, 2], [5, 3, 2] and [4, 4, 2]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then there are exactly k-1 parts equal to k. Example: a(10)=4 because we have [3, 3, 2, 2], [3, 3, 2, 1, 1], [3, 3, 1, 1, 1, 1] and [2, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then k occurs at least k+1 times. Example: a(10)=4 because we have [2, 2, 2, 2, 2], [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that the smallest part is larger than the number of parts. Example: a(10)=4 because we have [10], [7, 3], [6, 4] and [5, 5]. - Emeric Deutsch, Apr 09 2006
Also number of partitions into distinct parts where parts differ by at least 2 and with minimal part >= 2, a(0)=1 because the condition is void for the empty list. - Joerg Arndt, Jan 04 2011
The g.f. is the special case D=2 of Sum_{n>=0} x^(D*n*(n+1)/2) / Product_{k=1..n} (1-x^k), the g.f. or partitions into distinct parts where the difference between successive parts is >= D and the minimal part >= D. - Joerg Arndt, Mar 31 2014
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[2](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109699 and A109698. - Vaclav Kotesovec, Jan 21 2017

Examples

			G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + 4*x^11 + ...
G.f. = q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + 3*q^551 + ...
From _Joerg Arndt_, Dec 27 2012: (Start)
The a(18)=15: the partitions of 18 where all parts are 2 or 3 (mod 5) are
[ 1]  [ 2 2 2 2 2 2 2 2 2 ]
[ 2]  [ 3 3 2 2 2 2 2 2 ]
[ 3]  [ 3 3 3 3 2 2 2 ]
[ 4]  [ 3 3 3 3 3 3 ]
[ 5]  [ 7 3 2 2 2 2 ]
[ 6]  [ 7 3 3 3 2 ]
[ 7]  [ 7 7 2 2 ]
[ 8]  [ 8 2 2 2 2 2 ]
[ 9]  [ 8 3 3 2 2 ]
[10]  [ 8 7 3 ]
[11]  [ 8 8 2 ]
[12]  [ 12 2 2 2 ]
[13]  [ 12 3 3 ]
[14]  [ 13 3 2 ]
[15]  [ 18 ]
(End)
From _Wolfdieter Lang_, Oct 29 2016: (Start)
The a(18)=15 partitions of 18 without part 1 and parts differing by at least 2 are:
  [18]; [16,2], [15,3], [14,4], [13,5], [12,6], [11,7], [10,8]; [12,4,2], [11,5,2], [10,6,2], [9,7,2],[10,5,3], [9,6,3], [8,6,4]. The semicolon separates different number of parts. The maximal number of parts is A259361(18) = 3. (End)

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.
  • G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(f), p. 591.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 290-291.
  • H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003114.
Cf. A047221, A219607.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.

Programs

  • Haskell
    a003106 = p a047221_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, Nov 30 2012
  • Maple
    g:=1/product((1-x^(5*j-2))*(1-x^(5*j-3)),j=1..15): gser:=series(g,x=0,66): seq(coeff(gser,x,n),n=0..63); # Emeric Deutsch, Apr 09 2006
  • Mathematica
    max = 63; g[x_] := 1/Product[(1-x^(5j-2))*(1-x^(5j-3)), {j, 1, Floor[max/4]}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 17 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] > Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, -1, -1, 0, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
nmax = 63; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 2}~Join~{Range[0, kmax]*5 + 3}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)
  • PARI
    {a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, (sqrtint(4*n + 1) - 1) \ 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n - k^2 - k))), 1), n))}; /* Michael Somos, Oct 15 2008 */

Formula

The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n*(n+1))/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-2))*(1-t^(5*n-3))); this is the g.f. for the sequence.
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2 + 2*k) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - Michael Somos, Oct 15 2008
From Joerg Arndt, Oct 10 2012: (Start)
Bill Gosper gives (message to the math-fun mailing list, Oct 07 2012)
prod(k>=0, [0 , a; q^k, 1]) = [0, X(a,q); 0, Y(a,q)] where
X(a,q) = a * sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^n) ) and
Y(a,q) = sum(n>=0, a^n*q^(n^2-n) / prod(k=1..n, 1-q^n) ).
Set a=q to obtain prod(k>=0, [0 , a; q^k, 1]) = [0, q*H(q); 0, G(q)] where
H(q) is the g.f. of A003106 and G(q) is the g.f. of A003114.
Bill Gosper and N. J. A. Sloane give (message to math-fun, Oct 10 2012)
prod(k>=0, [0 , a*q^k; 1, 1]) = [U(a,q), U(a,q); V(a,q), V(a,q)] where
U(a,q) = a * sum(n>=0, a^n*q^(n^2+n) / prod(k=1..n, 1-q^k) ) and
V(a,q) = sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^k) ).
Set a=1 to obtain prod(k>=0, [0 , q^k; 1, 1]) = [H(q), H(q); G(q), G(q)].
(End)
Expansion of f(-x^5) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, May 06 2015
Expansion of f(-x, -x^4) / f(-x) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ sqrt((sqrt(5)-1)/5) * exp(2*Pi*sqrt(n/15)) / (2^(3/2) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(60*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 24 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284152(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

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