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 A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4. (Formerly M0566 N0204) 66
 1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, 105, 132, 166, 208, 258, 320, 395, 484, 592, 722, 876, 1060, 1280, 1539, 1846, 2210, 2636, 3138, 3728, 4416, 5222, 6163, 7256, 8528, 10006, 11716, 13696, 15986, 18624, 21666, 25169, 29190, 33808, 39104, 45164 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also number of partitions of n where no part appears more than three times. a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.). Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006 Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010 Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010 The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015 Convolution inverse is A082303. - Michael Somos, Sep 30 2017 REFERENCES A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128. I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2). M. D. Hirschhorn, The Power of q, Springer, 2017. See ped page 303ff. R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe) George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 9.) George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See Theorem 1.4 p. 75. Cristina Ballantine and Mircea Merca, Parity of sums of partition numbers and squares in arithmetic progressions, The Ramanujan Journal, 2016. A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129.] S.-C. Chen, On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943. A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008). M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014). Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15. Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12. Alexander Patkowski, On some partitions where even parts do not repeat, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263. Eric Weisstein's World of Mathematics, Partition Function b_k and Partition Function P. Wikipedia, Glaisher's Theorem. FORMULA Euler transform of period 4 sequence [ 1, 1, 1, 0, ...]. Expansion of q^(-1/8) * eta(q^4) / eta(q) in powers of q. - Michael Somos, Mar 19 2004 Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 08 2011 G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - Jon Perry, Mar 30 2004 G.f.: Product_{k>=1} (1+x^k)^(2-k%2). - Jon Perry, May 05 2005 G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)). G.f.: Sum_{n>=0} ( x^(n*(n+1)/2) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - Joerg Arndt, Apr 07 2011 G.f.: P(x^4)/P(x) where P(x) = Product_{k>=1} 1-x^k. - Joerg Arndt, Jun 21 2011 A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n). G.f.: (1+ 1/G(0))/2, where G(k) = 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013 G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - Paul D. Hanna, Jul 24 2013 a(n) ~ Pi * BesselI(1, sqrt(8*n + 1)*Pi/4) / (2*sqrt(8*n + 1)) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)) * (1 + (Pi/(16*sqrt(2)) - 3/(4*Pi*sqrt(2))) / sqrt(n) + (Pi^2/1024 - 15/(64*Pi^2) - 15/128) / n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 14 2017 a(n) = (1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017 G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082303. - Michael Somos, Sep 30 2017 EXAMPLE G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ... G.f. = q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ... a(5)=6 because we have , [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1]. MAPLE g:=product((1+x^j)*(1+x^(2*j)), j=1..50): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..48); # Emeric Deutsch, Apr 19 2006 # second Maple program: with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(      `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)     end: seq(a(n), n=0..50);  # Alois P. Heinz, Nov 24 2015 MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}]; (* Michael Somos, Jul 11 2011 *) a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 08 2011 *) CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j, 1, 48}], {x, 0, 48}], x] (* Jean-François Alcover, May 26 2011, after Jon Perry *) QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* Jean-François Alcover, Nov 24 2015 *) a = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *) Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 49}] (* Robert Price, Jul 28 2020 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)), n))}; (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))}; /* Michael Somos, Jun 01 2004 */ (Haskell) a001935 = p a042968_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, Sep 02 2012 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)), n)} \\ Paul D. Hanna, Jul 24 2013 CROSSREFS Cf. A000009, A000726, A001936, A035959, A035985, A042968, A061198, A061199, A070048, A081055, A081056, A083365, A098491, A098492, A219601. Cf. A000041, A010054. - Gary W. Adamson, Mar 16 2010 Cf. A174715. - Gary W. Adamson, Mar 27 2010 Cf. A082303. Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546. Sequence in context: A069907 A280424 A083365 * A286141 A007604 A013950 Adjacent sequences:  A001932 A001933 A001934 * A001936 A001937 A001938 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from James A. Sellers STATUS approved

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