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%I M0261 #138 Aug 23 2023 08:35:02
%S 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,
%T 58,64,75,82,95,105,120,133,152,167,190,210,237,261,295,324,364,401,
%U 448,493,551,604,673,739,820,899,997,1091,1207,1321,1457,1593,1756,1916,2108,2301
%N Number of partitions of n into parts 5k+2 or 5k+3.
%C Expansion of Rogers-Ramanujan function H(x) in powers of x.
%C 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
%C Example: a(10)=4 because we have [9, 1], [6, 2, 2], [5, 3, 2] and [4, 4, 2]. - _Emeric Deutsch_, Apr 09 2006
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%C Convolution of A109699 and A109698. - _Vaclav Kotesovec_, Jan 21 2017
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.
%D G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(f), p. 591.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 290-291.
%D H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A003106/b003106.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
%H George E. Andrews; R. J. Baxter, <a href="http://www.computing-wisdom.com/jstor/rogers-ramanujan.pdf">A motivated proof of the Rogers-Ramanujan identities</a>, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
%H R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/hep-th/0505097">Parafermionic derivation of Andrews-type multiple-sums</a>, arXiv:hep-th/0505097, 2005.
%H James Lepowsky and Minxian Zhu, <a href="http://arxiv.org/abs/1205.6570">A motivated proof of Gordon's identities</a>, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
%H I. Martinjak, D. Svrtan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Martinjak/mart3.html">New Identities for the Polarized Partitions and Partitions with d-Distant Parts</a>, J. Int. Seq. 17 (2014) # 14.11.4.
%H Herman P. Robinson, <a href="/A003105/a003105.pdf">Letter to N. J. A. Sloane, Jan 1974</a>.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanIdentities.html">Rogers-Ramanujan Identities</a>
%F 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.
%F 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
%F Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - _Michael Somos_, Oct 15 2008
%F From _Joerg Arndt_, Oct 10 2012: (Start)
%F _Bill Gosper_ gives (message to the math-fun mailing list, Oct 07 2012)
%F prod(k>=0, [0 , a; q^k, 1]) = [0, X(a,q); 0, Y(a,q)] where
%F X(a,q) = a * sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^n) ) and
%F Y(a,q) = sum(n>=0, a^n*q^(n^2-n) / prod(k=1..n, 1-q^n) ).
%F Set a=q to obtain prod(k>=0, [0 , a; q^k, 1]) = [0, q*H(q); 0, G(q)] where
%F H(q) is the g.f. of A003106 and G(q) is the g.f. of A003114.
%F _Bill Gosper_ and _N. J. A. Sloane_ give (message to math-fun, Oct 10 2012)
%F prod(k>=0, [0 , a*q^k; 1, 1]) = [U(a,q), U(a,q); V(a,q), V(a,q)] where
%F U(a,q) = a * sum(n>=0, a^n*q^(n^2+n) / prod(k=1..n, 1-q^k) ) and
%F V(a,q) = sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^k) ).
%F Set a=1 to obtain prod(k>=0, [0 , q^k; 1, 1]) = [H(q), H(q); G(q), G(q)].
%F (End)
%F 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
%F 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
%F 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
%F a(n) = (1/n)*Sum_{k=1..n} A284152(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 21 2017
%e 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 + ...
%e G.f. = q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + 3*q^551 + ...
%e From _Joerg Arndt_, Dec 27 2012: (Start)
%e The a(18)=15: the partitions of 18 where all parts are 2 or 3 (mod 5) are
%e [ 1] [ 2 2 2 2 2 2 2 2 2 ]
%e [ 2] [ 3 3 2 2 2 2 2 2 ]
%e [ 3] [ 3 3 3 3 2 2 2 ]
%e [ 4] [ 3 3 3 3 3 3 ]
%e [ 5] [ 7 3 2 2 2 2 ]
%e [ 6] [ 7 3 3 3 2 ]
%e [ 7] [ 7 7 2 2 ]
%e [ 8] [ 8 2 2 2 2 2 ]
%e [ 9] [ 8 3 3 2 2 ]
%e [10] [ 8 7 3 ]
%e [11] [ 8 8 2 ]
%e [12] [ 12 2 2 2 ]
%e [13] [ 12 3 3 ]
%e [14] [ 13 3 2 ]
%e [15] [ 18 ]
%e (End)
%e From _Wolfdieter Lang_, Oct 29 2016: (Start)
%e The a(18)=15 partitions of 18 without part 1 and parts differing by at least 2 are:
%e [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)
%p 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
%t 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_ *)
%t Table[Count[IntegerPartitions[n], p_ /; Min[p] > Length[p]], {n, 40}] (* _Clark Kimberling_, Feb 13 2014 *)
%t a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* _Michael Somos_, May 06 2015 *)
%t 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 *)
%t nmax = 63; kmax = nmax/5;
%t s = Flatten[{Range[0, kmax]*5 + 2}~Join~{Range[0, kmax]*5 + 3}];
%t Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Jul 31 2020 *)
%o (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 */
%o (Haskell)
%o a003106 = p a047221_list where
%o p _ 0 = 1
%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o -- _Reinhard Zumkeller_, Nov 30 2012
%Y Cf. A003114.
%Y Cf. A047221, A219607.
%Y 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.
%K nonn,nice,easy
%O 0,7
%A _N. J. A. Sloane_, _Herman P. Robinson_
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