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 A003114 Number of partitions of n into parts 5k+1 or 5k+4. (Formerly M0266) 111
 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 61, 70, 79, 91, 102, 117, 131, 149, 167, 189, 211, 239, 266, 299, 333, 374, 415, 465, 515, 575, 637, 709, 783, 871, 961, 1065, 1174, 1299, 1429, 1579, 1735, 1913, 2100, 2311, 2533, 2785 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Expansion of Rogers-Ramanujan function G(x) in powers of x. Same as number of partitions into distinct parts where the difference between successive parts is >= 2. As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001 The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))). Coefficients in expansion of permanent of infinite tridiagonal matrix: 1 1 0 0 0 0 0 0 ... x 1 1 0 0 0 0 0 ... 0 x^2 1 1 0 0 0 ... 0 0 x^3 1 1 0 0 ... 0 0 0 x^4 1 1 0 ... ................... - Vladeta Jovovic, Jul 17 2004 Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Jul 17 2004 Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Feb 27 2006 Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 16 2006 a(n) = number of NW partitions of n, for n >= 1; see A237981. For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G(x). - N. J. A. Sloane, Nov 22 2015 Convolution of A109700 and A109697. - Vaclav Kotesovec, Jan 21 2017 REFERENCES G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238. G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(e), p. 591. Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107. G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 90-92. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth 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 Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe) G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p. G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409. R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986. R. K. Guy, Letter to N. J. A. Sloane, 1987 R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy] P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums, arXiv:hep-th/0505097, 2005. James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211. I. Martinjak, D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4. Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974. A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp. Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities. FORMULA G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i). The g.f. above is the special case D=2 of sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ), the g.f. for partitions into distinct part where the difference between successive parts is >= D. - Joerg Arndt, Mar 31 2014 G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)). - Jon Perry, Jul 06 2004 G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006 Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - Michael Somos, Oct 15 2008 Expansion of f(-x^5) / f(-x^1, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, May 17 2015 Expansion of f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015 a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * 5^(1/2) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(60*sqrt(15))) / sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 23 2015, extended Jan 24 2017 a(n) = (1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017 EXAMPLE G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ... G.f. = 1/q + q^59 + q^119 + q^179 + 2*q^239 + 2*q^299 + 3*q^359 + 3*q^419 + ... From Joerg Arndt, Dec 27 2012: (Start) The a(16)=17 partitions of 16 where all parts are 1 or 4 (mod 5) are [ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] [ 2]  [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ] [ 3]  [ 4 4 1 1 1 1 1 1 1 1 ] [ 4]  [ 4 4 4 1 1 1 1 ] [ 5]  [ 4 4 4 4 ] [ 6]  [ 6 1 1 1 1 1 1 1 1 1 1 ] [ 7]  [ 6 4 1 1 1 1 1 1 ] [ 8]  [ 6 4 4 1 1 ] [ 9]  [ 6 6 1 1 1 1 ]   [ 6 6 4 ]   [ 9 1 1 1 1 1 1 1 ]   [ 9 4 1 1 1 ]   [ 9 6 1 ]   [ 11 1 1 1 1 1 ]   [ 11 4 1 ]   [ 14 1 1 ]   [ 16 ] The a(16)=17 partitions of 16 where successive parts differ by at least 2 are [ 1]  [ 7 5 3 1 ] [ 2]  [ 8 5 3 ] [ 3]  [ 8 6 2 ] [ 4]  [ 9 5 2 ] [ 5]  [ 9 6 1 ] [ 6]  [ 9 7 ] [ 7]  [ 10 4 2 ] [ 8]  [ 10 5 1 ] [ 9]  [ 10 6 ]   [ 11 4 1 ]   [ 11 5 ]   [ 12 3 1 ]   [ 12 4 ]   [ 13 3 ]   [ 14 2 ]   [ 15 1 ]   [ 16 ] (End) MAPLE g:=sum(x^(k^2)/product(1-x^j, j=1..k), k=0..10): gser:=series(g, x=0, 65): seq(coeff(gser, x, n), n=0..60); # Emeric Deutsch, Feb 27 2006 MATHEMATICA CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* Jean-François Alcover, Apr 08 2011, after Emeric Deutsch *) Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 24}] (* _Clark Kimberling, Feb 13 2014 *) a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^5]  QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, May 17 2015 *) a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 0, 0, -1, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *) PROG (PARI) {a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1), n))}; /* Michael Somos, Oct 15 2008 */ (Haskell) a003114 = p a047209_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, Jan 05 2011 (Haskell) a003114 = p 1 where    p _ 0 = 1    p k m = if k > m then 0 else p (k + 2) (m - k) + p (k + 1) m -- Reinhard Zumkeller, Feb 19 2013 CROSSREFS Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900. Cf. A188216 (least part k occurs at least k times). Cf. A047209, A203776, A237981. For the generalized Rogers-Ramanujan series G, G, G, G, G, G, G, G see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G = G+G is given by A003113. Row sums of A268187. Sequence in context: A000607 A114372 A046676 * A185227 A217569 A026823 Adjacent sequences:  A003111 A003112 A003113 * A003115 A003116 A003117 KEYWORD easy,nonn,nice AUTHOR STATUS approved

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Last modified May 21 00:54 EDT 2019. Contains 323428 sequences. (Running on oeis4.)