A203776 -id:A203776 - cp's OEIS frontend

A003114 Number of partitions of n into parts 5k+1 or 5k+4.

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

Offset: 0

N. J. A. Sloane and Herman P. Robinson

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[1](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109700 and A109697. - Vaclav Kotesovec, Jan 21 2017

			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 ]
  [10]  [ 6 6 4 ]
  [11]  [ 9 1 1 1 1 1 1 1 ]
  [12]  [ 9 4 1 1 1 ]
  [13]  [ 9 6 1 ]
  [14]  [ 11 1 1 1 1 1 ]
  [15]  [ 11 4 1 ]
  [16]  [ 14 1 1 ]
  [17]  [ 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 ]
  [10]  [ 11 4 1 ]
  [11]  [ 11 5 ]
  [12]  [ 12 3 1 ]
  [13]  [ 12 4 ]
  [14]  [ 13 3 ]
  [15]  [ 14 2 ]
  [16]  [ 15 1 ]
  [17]  [ 16 ]
(End)

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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
George 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.
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 14.
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities.
Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

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[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Row sums of A268187.

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

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

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

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, 0, 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 *)
nmax = 60; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 1}~Join~{Range[0, kmax]*5 + 4}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

{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 */

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

A047209 Numbers that are congruent to {1, 4} mod 5.

Original entry on oeis.org

1, 4, 6, 9, 11, 14, 16, 19, 21, 24, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 54, 56, 59, 61, 64, 66, 69, 71, 74, 76, 79, 81, 84, 86, 89, 91, 94, 96, 99, 101, 104, 106, 109, 111, 114, 116, 119, 121, 124, 126, 129, 131, 134, 136, 139, 141, 144, 146, 149, 151, 154

Offset: 1

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 72 ).
Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in our case, a(n)^2 - 1 == 0 (mod 5). - Bruno Berselli, Nov 17 2010
The sum of the alternating series (-1)^(n+1)/a(n) from n=1 to infinity is (Pi/5)*cot(Pi/5), that is (1/5)*sqrt(1 + 2/sqrt(5))*Pi. - Jean-François Alcover, May 03 2013
These numbers appear in the product of a Rogers-Ramanujan identity. See A003114 also for references. - Wolfdieter Lang, Oct 29 2016
Let m be a product of any number of terms of this sequence. Then m - 1 or m + 1 is divisible by 5. Closed under multiplication. - David A. Corneth, May 11 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
William A. Stein, The modular forms database.
Eric Weisstein's World of Mathematics, Determined by Spectrum.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000566, A036666, A001622, A003114, A203776, A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887, A352324.
Cf. A005408 (n=1 or 3 mod 4), A007310 (n=1 or 5 mod 6).
Cf. A045468 (primes), A032527 (partial sums).

a047209 = (flip div 2) . (subtract 2) . (* 5)
a047209_list = 1 : 4 : (map (+ 5) a047209_list)
-- Reinhard Zumkeller, Jul 19 2013, Jan 05 2011

seq(floor(5*k/2)-1, k=1..100); # Wesley Ivan Hurt, Sep 27 2013

Select[Range[0, 200], MemberQ[{1, 4}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)
LinearRecurrence[{1,1,-1},{1,4,6},70] (* Harvey P. Dale, Jul 19 2024 *)

a(n)=(10*n+(-1)^n-5)/4 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1+3x+x^2)/((1-x)(1-x^2)).
a(n) = floor((5n-2)/2). [corrected by Reinhard Zumkeller, Jul 19 2013]
a(1) = 1, a(n) = 5(n-1) - a(n-1). - Benoit Cloitre, Apr 12 2003
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = (10*n + (-1)^n - 5)/4.
a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.
a(n) = a(n-2) + 5 for n > 2.
a(n) = 5*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
a(n)^2 = 5*A036666(n) + 1 (cf. also Comments). (End)
a(n) = 5*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011
E.g.f.: 1 + ((10*x - 5)*exp(x) + exp(-x))/4. - David Lovler, Aug 23 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = phi (A001622).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/5) * cosec(Pi/5) (A352324). (End)

Edited by Michael Somos, Sep 22 2002

A281243 Expansion of Product_{k>=1} (1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 1, 0, 0, 2, 3, 1, 0, 0, 3, 4, 1, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 2, 7, 5, 1, 0, 3, 8, 6, 1, 0, 5, 10, 6, 1, 1, 6, 12, 7, 1, 1, 9, 14, 7, 1, 2, 11, 16, 8, 1

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Convolution of this sequence and A280454 is A203776.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A262928, A147599, A281244, A281245.

Cf. A261612, A169975, A280454, A280456, A280457.

nmax = 200; CoefficientList[Series[Product[(1 + x^(5*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 5] == 4, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/15)*Pi) / (2^(9/5)*15^(1/4)*n^(3/4)) * (1 + (Pi/(240*sqrt(15)) - 3*sqrt(15)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

G.f.: Sum_{k>=0} x^(k*(5*k + 3)/2) / Product_{j=1..k} (1 - x^(5*j)). - Ilya Gutkovskiy, Nov 24 2020

A280454 Expansion of Product_{k>=0} (1 + x^(5*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 1, 3, 3, 1, 0, 1, 4, 4, 1, 0, 1, 4, 5, 2, 0, 1, 5, 7, 3, 0, 1, 5, 8, 5, 1, 1, 6, 10, 6, 1, 1, 6, 12, 9, 2, 1, 7, 14, 11, 3, 1, 7, 16, 15, 5, 1, 8, 19, 18, 7, 2, 8, 21, 23, 10, 2, 9, 24, 27, 13, 3, 9, 27, 34, 18, 4, 10, 30, 39, 23, 6, 10

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 5.

Convolution of A281243 and A280454 is A203776. - Vaclav Kotesovec, Jan 18 2017

			a(27) = 3 because we have [26, 1], [21, 6] and [11, 16].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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.
Index entries for related partition-counting sequences

Cf. A000700, A016861, A109697, A169975, A261612, A280456, A280457.

Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 102; CoefficientList[Series[Product[(1 + x^(5 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 5] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(5*k+1)).

a(n) ~ exp(Pi*sqrt(n)/sqrt(15))/(2*2^(1/5)*15^(1/4)*n^(3/4)) * (1 + (Pi/(240*sqrt(15)) - 3*sqrt(15)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017

A219607 Number of partitions of n into distinct parts 5k+2 or 5k+3.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 1, 2, 2, 1, 3, 1, 3, 3, 2, 5, 3, 5, 5, 4, 7, 4, 7, 7, 6, 11, 7, 11, 11, 9, 15, 10, 15, 16, 14, 22, 16, 23, 23, 20, 30, 22, 31, 32, 29, 42, 33, 44, 45, 41, 56, 45, 59, 61, 57, 78, 64, 82, 84, 78, 103, 86, 108, 112, 107, 138

Offset: 0

Reinhard Zumkeller, Nov 30 2012

nonn

Convolution of A281271 and A281272. - Vaclav Kotesovec, Jan 18 2017

			a(10) = #{8+2, 7+3} = 2;
a(11) = #{8+3} = 1;
a(12) = #{12, 7+3+2} = 2;
a(13) = #{13, 8+3+2} = 2;
a(14) = #{12+2} = 1;
a(15) = #{13+2, 12+3, 8+7} = 3;
a(16) = #{13+3} = 1;
a(17) = #{17, 12+3+2, 8+7+2} = 3;
a(18) = #{18, 13+3+2, 8+7+3} = 3;
a(19) = #{17+2, 12+7} = 2;
a(20) = #{18+2, 17+3, 13+7, 12+8, 8+7+3+2} = 5.

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..250 from Reinhard Zumkeller)

Cf. A047221, A003106, A203776.

a219607 = p a047221_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

nmax = 100; CoefficientList[Series[Product[(1 + x^(5*k - 2))*(1 + x^(5*k - 3)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 18 2017 *)

a(n) ~ exp(sqrt(2*n/15)*Pi) / (2*30^(1/4)*n^(3/4)) * (1 - (3*sqrt(15/2)/(8*Pi) + 11*Pi/(60*sqrt(30))) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

A301562 Expansion of Product_{k>=0} (1 + x^(5k+1))(1 + x^(5*k+2)).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 2, 3, 3, 2, 2, 3, 5, 6, 5, 4, 4, 6, 8, 8, 7, 7, 9, 12, 13, 11, 10, 12, 16, 19, 19, 17, 18, 23, 27, 27, 25, 25, 30, 37, 40, 38, 37, 42, 50, 55, 54, 52, 57, 68, 77, 78, 75, 78, 90, 102, 106, 104, 106, 120, 138, 146, 144, 145, 158

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

Number of partitions of n into distinct parts congruent to 1 or 2 mod 5.

			a(13) = 3 because we have [12, 1], [11, 2] and [7, 6].

Index entries for sequences related to partitions

Cf. A035371, A047216, A107234, A107235, A203776, A219607, A280454, A281272, A301563, A301564, A301565, A301567, A301568, A301569, A301570.

nmax = 70; CoefficientList[Series[Product[(1 + x^(5 k + 1)) (1 + x^(5 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x, x^5] QPochhammer[-x^2, x^5], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 2}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A047216(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(17/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301563 Expansion of Product_{k>=0} (1 + x^(5k+1))(1 + x^(5*k+3)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 4, 3, 4, 4, 4, 6, 4, 6, 7, 5, 9, 8, 8, 11, 9, 12, 12, 12, 16, 13, 17, 19, 17, 23, 21, 24, 27, 24, 32, 30, 32, 40, 35, 43, 45, 44, 53, 50, 59, 62, 61, 75, 70, 78, 87, 83, 99, 97, 105, 118, 112, 133, 134, 138, 159, 153

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

Number of partitions of n into distinct parts congruent to 1 or 3 mod 5.

			a(14) = 3 because we have [13, 1], [11, 3] and [8, 6].

Index entries for sequences related to partitions

Cf. A035372, A047219, A107234, A107236, A203776, A219607, A280454, A281271, A301562, A301564, A301565, A301567, A301568, A301569, A301570.

nmax = 72; CoefficientList[Series[Product[(1 + x^(5 k + 1)) (1 + x^(5 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 72; CoefficientList[Series[QPochhammer[-x, x^5] QPochhammer[-x^3, x^5], {x, 0, nmax}], x]
nmax = 72; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 3}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A047219(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(21/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301564 Expansion of Product_{k>=0} (1 + x^(5k+2))(1 + x^(5*k+4)).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 2, 1, 2, 2, 1, 3, 1, 3, 3, 2, 5, 2, 6, 3, 5, 6, 4, 8, 5, 8, 8, 7, 12, 7, 13, 11, 11, 16, 11, 19, 14, 19, 21, 17, 27, 20, 27, 28, 26, 36, 28, 40, 37, 38, 49, 39, 55, 49, 55, 64, 55, 76, 65, 78, 84, 78, 100, 87, 107, 109, 107, 134, 116, 145

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

Number of partitions of n into distinct parts congruent to 2 or 4 mod 5.

			a(16) = 3 because we have [14, 2], [12, 4] and [9, 7].

Index entries for sequences related to partitions

Cf. A035373, A047211, A107235, A107237, A203776, A219607, A281243, A281272, A301562, A301563, A301565, A301567, A301568, A301569, A301570.

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k + 2)) (1 + x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[QPochhammer[-x^2, x^5] QPochhammer[-x^4, x^5], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{2, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A047211(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(29/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301565 Expansion of Product_{k>=0} (1 + x^(5k+3))(1 + x^(5*k+4)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 4, 4, 3, 3, 4, 6, 6, 4, 4, 7, 9, 7, 6, 8, 11, 12, 10, 9, 12, 16, 16, 14, 14, 19, 23, 22, 19, 21, 27, 31, 29, 26, 31, 40, 42, 38, 38, 45, 53, 55, 51, 52, 63, 73, 73, 69, 73, 87, 97, 95, 91, 100, 118, 128

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

Number of partitions of n into distinct parts congruent to 3 or 4 mod 5.

			a(17) = 3 because we have [14, 3], [13, 4] and [9, 8].

Index entries for sequences related to partitions

Cf. A035374, A047204, A107236, A107237, A203776, A219607, A281243, A281271, A301562, A301563, A301564, A301567, A301568, A301569, A301570.

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k + 3)) (1 + x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[QPochhammer[-x^3, x^5] QPochhammer[-x^4, x^5], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{3, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A047204(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(33/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301567 Expansion of Product_{k>=1} (1 + x^(5k))(1 + x^(5*k-4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 4, 1, 0, 2, 7, 7, 2, 0, 3, 10, 11, 4, 0, 4, 14, 17, 8, 1, 5, 19, 25, 13, 2, 6, 25, 36, 21, 4, 8, 33, 50, 33, 8, 10, 43, 69, 49, 14, 13, 55, 93, 71, 23, 17, 70, 124, 102, 37, 22, 88, 163, 142, 57, 30, 110, 212, 195, 85

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 1 mod 5.

			a(11) = 3 because we have [11], [10, 1] and [6, 5].

Index entries for sequences related to partitions

Cf. A008851, A035367, A036820, A203776, A219607, A280454, A301562, A301563, A301564, A301565, A301568, A301569, A301570.

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 4)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[x^4 QPochhammer[-1, x^5] QPochhammer[-x^(-4), x^5]/(2 (1 + x^4)), {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 1}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^A008851(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(29/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

cp's OEIS Frontend

A003114 Number of partitions of n into parts 5k+1 or 5k+4.

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A047209 Numbers that are congruent to {1, 4} mod 5.

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A281243 Expansion of Product_{k>=1} (1 + x^(5*k-1)).

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A280454 Expansion of Product_{k>=0} (1 + x^(5*k+1)).

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A219607 Number of partitions of n into distinct parts 5*k+2 or 5*k+3.

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A301562 Expansion of Product_{k>=0} (1 + x^(5*k+1))*(1 + x^(5*k+2)).

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A301563 Expansion of Product_{k>=0} (1 + x^(5*k+1))*(1 + x^(5*k+3)).

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A219607 Number of partitions of n into distinct parts 5k+2 or 5k+3.

A301562 Expansion of Product_{k>=0} (1 + x^(5k+1))(1 + x^(5*k+2)).

A301563 Expansion of Product_{k>=0} (1 + x^(5k+1))(1 + x^(5*k+3)).

A301564 Expansion of Product_{k>=0} (1 + x^(5k+2))(1 + x^(5*k+4)).

A301565 Expansion of Product_{k>=0} (1 + x^(5k+3))(1 + x^(5*k+4)).

A301567 Expansion of Product_{k>=1} (1 + x^(5k))(1 + x^(5*k-4)).