A035959 Number of partitions of n in which no parts are multiples of 5.

1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, 127, 164, 205, 262, 325, 409, 505, 628, 769, 950, 1156, 1414, 1713, 2081, 2505, 3026, 3625, 4352, 5192, 6200, 7364, 8756, 10357, 12258, 14450, 17034, 20006, 23500, 27510, 32200, 37582, 43846, 51022

Offset

0,3

Comments

Also number of partitions with at most 4 parts of size 1 and differences between parts at distance 6 are greater than 1.

Also number of partitions of n where no part appears more than four times.

Case k=7, i=5 of Gordon Theorem.

References

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.

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.

G. N. Watson, Ramanujans Vermutung ueber Zerfaellungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression Y = C/B in the notation of p. 106. [Added by N. J. A. Sloane, Nov 13 2009]

Eric Weisstein's World of Mathematics, Partition Function b_k.

Wikipedia, Glaisher's Theorem

Formula

G.f.: Product_{j>=1} (1 + x^j + x^2j + x^3j + x^4j). - Jon Perry, Mar 30 2004

G.f.: Product_{n>0, n==1, 2, 3, 4 mod 5} 1/(1-q^n).

Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - u*v - 5*u^2*v^2. - Michael Somos, May 28 2006

Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v + 5*v^2*(u + w) - (u^2 + u*w + w^2). - Michael Somos, May 28 2006

Euler transform of period 5 sequence [ 1, 1, 1, 1, 0, ...]. - Michael Somos, May 28 2006

G.f.: Product_{k > 0} P5(x^k) where P5 is 5th cyclotomic polynomial.

Convolution inverse is A145466. - Michael Somos, Jun 26 2014

a(n) ~ 2*Pi * BesselI(1, 2*sqrt((6*n + 1)/5) * Pi/3) / (5*sqrt(6*n + 1)) ~ exp(2*Pi*sqrt(2*n/15)) / (3^(1/4) * 10^(3/4) * n^(3/4)) * (1 + (Pi/(3*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(2*n) + (Pi^2/540 - 225/(1024*Pi^2) - 5/32) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017

a(n) = (1/n)*Sum_{k=1..n} A116073(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k))/(k*(1 - x^(5*k)))). - Ilya Gutkovskiy, Aug 15 2018

Example

G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 13*x^7 + 19*x^8 + ...
G.f. = q + q^7 + 2*q^13 + 3*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 13*q^43 + ...
a(6) counts these partitions: 6, 42, 411, 33, 321, 3111, 2211, 21111, 111111. - Clark Kimberling, Mar 09 2014

Mathematica

max = 47; f[x_] := (x^5-1)/(x-1); g[x_] := Product[f[x^k], {k, 1, max}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 29 2011, after Michael Somos *)
t = Flatten[Table[5 n + r, {n, 0, 60}, {r, 1, 4}]]; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 8}] (* shows partitions *)
a[n_] := Length@p@n; a /@ Range[0, 50] (* Clark Kimberling, Mar 09 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^5]/QP[q] + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 5], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)
Table[Count[IntegerPartitions[n], _?(NoneTrue[Mod[#, 5]==0&])], {n, 0, 50}] (* Harvey P. Dale, Dec 25 2021 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) / eta(x + x * O(x^n)), n))}; /* Michael Somos, May 28 2006 */
(Haskell)
a035959 = p a047201_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, Dec 17 2011

Crossrefs

Cf. A061198, A061199, A047201.

Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10).

Cf. A096938, A133563, A320607.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Sequence in context: A341131 A288253 A341154 * A036801 A035966 A035974

Adjacent sequences: A035956 A035957 A035958 * A035960 A035961 A035962

Keyword

nonn,easy,nice

Author

Olivier Gérard

Status

approved