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 A078408 Number of ways to partition 2n+1 into distinct positive integers. 37
 1, 2, 3, 5, 8, 12, 18, 27, 38, 54, 76, 104, 142, 192, 256, 340, 448, 585, 760, 982, 1260, 1610, 2048, 2590, 3264, 4097, 5120, 6378, 7917, 9792, 12076, 14848, 18200, 22250, 27130, 32992, 40026, 48446, 58499, 70488, 84756, 101698, 121792, 145578, 173682 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is also the number of partitions of 2n+1 in which all parts are odd, due to Euler's partition theorem. See A000009. - Wolfdieter Lang, Jul 08 2012 REFERENCES G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Reinhard Zumkeller) Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA a(n) = t(2*n+1, 0), t as defined in A079211. Euler transform of period 16 sequence [ 2, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 0, 2, 0, ...]. - Michael Somos, Mar 04 2003 a(n)=A000009(2n+1). G.f.: 1/[(1-x)(1-x^3)(1-x^5)...] - Jon Perry, May 27 2004 Expansion of f(x, x^7) / f(-x, -x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015 From Peter Bala, Feb 04 2021: (Start) G.f.: Sum_{n >= 0} x^n/Product_{k = 1..2*n+1} 1 - x^k. Replace q with q^2 and set t = q in Andrews, equation 2.2.5, p. 19, and then take the odd part of the series. G.f.: 1/(1 - x)*Sum_{n>=0} x^floor(3*n/2)/Product_{k=1..n} (1 - x^k). (End) EXAMPLE a(3) = 5 because 7 = 1+6 = 2+5 = 3+4 = 1+2+4 (partitions into distinct parts) and 7 = 1+1+5 = 1+3+3 = 1+1+1+1+3 = 1+1+1+1+1+1+1 (partitions into odd parts). [Wolfdieter Lang, Jul 08 2012] G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 18*x^6 + 27*x^7 + 38*x^8 + ... G.f. = q^25 + 2*q^73 + 3*q^121 + 5*q^169 + 8*q^217 + 12*q^265 + 18*q^313 + ... MAPLE G := 1/(1 - x)*add(x^floor(3*n/2)/mul(1 - x^k, k = 1..n), n = 0..50): S := series(G, x, 76): seq(coeff(S, x, j), j = 0..75); # Peter Bala, Feb 04 2021 MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / QPochhammer[ x], {x, 0, 2 n + 1}]; (* Michael Somos, Oct 06 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))}; (Haskell) import Data.MemoCombinators (memo2, integral) a078408 n = a078408_list !! n a078408_list = f 1 where    f x = (p' 1 x) : f (x + 2)    p' = memo2 integral integral p    p _ 0 = 1    p k m = if m < k then 0 else p' k (m - k) + p' (k + 2) m -- Reinhard Zumkeller, Nov 27 2015 CROSSREFS Cf. A035294, A078409, A078410. Cf. A005408, A000009. Sequence in context: A280278 A136275 A328170 * A007478 A014605 A232477 Adjacent sequences:  A078405 A078406 A078407 * A078409 A078410 A078411 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Dec 27 2002 EXTENSIONS More terms from Reinhard Zumkeller, Dec 28 2002 STATUS approved

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Last modified April 11 08:59 EDT 2021. Contains 342886 sequences. (Running on oeis4.)