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 A087897 Number of partitions of n into odd parts greater than 1. 22
 1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, 13, 15, 18, 20, 23, 27, 30, 34, 40, 44, 50, 58, 64, 73, 83, 92, 104, 118, 131, 147, 166, 184, 206, 232, 256, 286, 320, 354, 394, 439, 485, 538, 598, 660, 730, 809, 891, 984, 1088, 1196, 1318, 1454, 1596, 1756 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS Also number of partitions of n into distinct parts which are not powers of 2. Also number of partitions of n into distinct parts such that the two largest parts differ by 1. Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006 Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007 In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)). Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013 Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014 REFERENCES J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 C. Ballantine, M. Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1; doi. B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973. Howard D. Grossman, Problem 228, Mathematics Magazine, 28 (1955), p. 160. R. K. Guy, Two theorems on partitions, Math. Gaz., 42 (1958), 84-86. Math. Rev. 20 #3110. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^(-1/24) * (1 - q) * eta(q^2) / eta(q) in powers of q. Expansion of (1 - x) / chi(-x) in powers of x where chi() is a Ramanujan theta function. G.f.: 1 + x^3 + x^5*(1 + x) + x^7*(1 + x)*(1 + x^2) + x^9*(1 + x)*(1 + x^2)*(1 + x^3) + ... [Glaisher 1876]. - Michael Somos, Jun 20 2012 G.f.: Product_{k >= 1} 1/(1-x^(2*k+1)). G.f.: Product_{k >= 1, k not a power of 2} (1+x^k). G.f.: sum(x^(3k)/product(1-x^(2j), j=1..k), k=1..infinity). - Emeric Deutsch, Mar 30 2006 a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (8 * 3^(3/4) * n^(5/4)) * (1 - (15*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (169*Pi^2/13824 + 385/384 + 315/(128*Pi^2))/n). - Vaclav Kotesovec, Aug 30 2015, extended Nov 04 2016 EXAMPLE 1 + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + 3*x^13 + ... q + q^73 + q^121 + q^145 + q^169 + q^193 + 2*q^217 + 2*q^241 + 2*q^265 + ... a(10)=2 because we have [7,3] and [5,5]. From Joerg Arndt, Jun 11 2013: (Start) There are a(22)=13 symmetric unimodal compositions of 22+3=25 where the maximal part appears three times: 01:  [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ] 02:  [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ] 03:  [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ] 04:  [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ] 05:  [ 1 1 1 2 5 5 5 2 1 1 1 ] 06:  [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ] 07:  [ 1 1 3 5 5 5 3 1 1 ] 08:  [ 1 1 7 7 7 1 1 ] 09:  [ 1 2 2 5 5 5 2 2 1 ] 10:  [ 1 4 5 5 5 4 1 ] 11:  [ 2 2 2 2 3 3 3 2 2 2 2 ] 12:  [ 2 3 5 5 5 3 2 ] 13:  [ 2 7 7 7 2 ] (End) MAPLE To get 128 terms: t4 := mul((1+x^(2^n)), n=0..7); t5 := mul((1+x^k), k=1..128): t6 := series(t5/t4, x, 100); t7 := seriestolist(t6); # second Maple program: b:= proc(n, i) option remember; `if`(n=0, 1,       `if`(i<3, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))     end: a:= n-> b(n, n-1+irem(n, 2)): seq(a(n), n=0..80);  # Alois P. Heinz, Jun 11 2013 MATHEMATICA max = 65; f[x_] := Product[ 1/(1 - x^(2k+1)), {k, 1, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 16 2011, after Emeric Deutsch *) b[n_, i_] := b[n, i] = If[n==0, 1, If[i<3, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]] ]; a[n_] := b[n, n-1+Mod[n, 2]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 01 2015, after Alois P. Heinz *) Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* Vaclav Kotesovec, Dec 01 2015 *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - x) * eta(x^2 + A) / eta(x + A), n))} /* Michael Somos, Nov 13 2011 */ (Haskell) a087897 = p [3, 5..] where    p [] _ = 0    p _  0 = 1    p ks'@(k:ks) m | m < k     = 0                   | otherwise = p ks' (m - k) + p ks m -- Reinhard Zumkeller, Aug 12 2011 CROSSREFS Cf. A000009, A002865. Sequence in context: A090184 A174575 A029057 * A029056 A226503 A036847 Adjacent sequences:  A087894 A087895 A087896 * A087898 A087899 A087900 KEYWORD nonn AUTHOR N. J. A. Sloane, Dec 04 2003 STATUS approved

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Last modified December 9 09:21 EST 2019. Contains 329877 sequences. (Running on oeis4.)