cp's OEIS Frontend

A227135 Partitions with parts repeated at most twice and repetition only allowed if first part has an even index (first index = 1).

%I A227135 #47 Feb 23 2025 07:05:03
%S A227135 1,1,1,2,2,4,4,6,8,10,12,17,20,25,31,39,47,58,69,85,102,123,145,175,
%T A227135 207,246,290,343,401,473,551,646,751,875,1012,1177,1358,1570,1807,
%U A227135 2083,2389,2746,3140,3597,4106,4690,5337,6082,6907,7848,8895,10085,11404,12902,14561,16438,18520,20864,23460,26385,29619
%N A227135 Partitions with parts repeated at most twice and repetition only allowed if first part has an even index (first index = 1).
%H A227135 Alois P. Heinz, <a href="/A227135/b227135.txt">Table of n, a(n) for n = 0..10000</a>
%F A227135 Conjecture: A227134(n) + A227135(n) = A182372(n) for n>=0, see comment in A182372.
%F A227135 G.f.: 1/(1-x) + Sum_{n>=2} x^(A002620(n+2)-1) / Product_{k=1..n} (1-x^k), where A002620(n) = floor(n/2)*ceiling(n/2) forms the quarter-squares. - _Paul D. Hanna_, Jul 06 2013
%F A227135 a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = 2^(3/4) / (sqrt(5)*(1 + sqrt(5))^(3/2)) = 0.1291995618069... - _Vaclav Kotesovec_, May 28 2018, updated Mar 06 2020
%e A227135 G.f.: 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 8*x^8 +...
%e A227135 G.f.: 1/(1-x) + x^3/((1-x)*(1-x^2)) + x^5/((1-x)*(1-x^2)*(1-x^3)) + x^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + x^11/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) + x^15/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)) +...
%e A227135 There are a(13)=25 such partitions, displayed here as partitions into two sorts of parts (format P:S for sort:part) where the first sort is 1 and sorts oscillate:
%e A227135 01:  [ 1:1  2:0  2:1  3:0  5:1  ]
%e A227135 02:  [ 1:1  2:0  2:1  4:0  4:1  ]
%e A227135 03:  [ 1:1  2:0  2:1  8:0  ]
%e A227135 04:  [ 1:1  2:0  3:1  7:0  ]
%e A227135 05:  [ 1:1  2:0  4:1  6:0  ]
%e A227135 06:  [ 1:1  2:0 10:1  ]
%e A227135 07:  [ 1:1  3:0  3:1  6:0  ]
%e A227135 08:  [ 1:1  3:0  4:1  5:0  ]
%e A227135 09:  [ 1:1  3:0  9:1  ]
%e A227135 10:  [ 1:1  4:0  8:1  ]
%e A227135 11:  [ 1:1  5:0  7:1  ]
%e A227135 12:  [ 1:1  6:0  6:1  ]
%e A227135 13:  [ 1:1 12:0  ]
%e A227135 14:  [ 2:1  3:0  3:1  5:0  ]
%e A227135 15:  [ 2:1  3:0  8:1  ]
%e A227135 16:  [ 2:1  4:0  7:1  ]
%e A227135 17:  [ 2:1  5:0  6:1  ]
%e A227135 18:  [ 2:1 11:0  ]
%e A227135 19:  [ 3:1  4:0  6:1  ]
%e A227135 20:  [ 3:1  5:0  5:1  ]
%e A227135 21:  [ 3:1 10:0  ]
%e A227135 22:  [ 4:1  9:0  ]
%e A227135 23:  [ 5:1  8:0  ]
%e A227135 24:  [ 6:1  7:0  ]
%e A227135 25:  [13:1  ]
%p A227135 ## See A227134
%p A227135 # second Maple program:
%p A227135 b:= proc(n, i, t) option remember; `if`(n=0, 1-t,
%p A227135       `if`(i*(i+1)<n, 0, add(b(n-i*j, i-1,
%p A227135       irem(t+j, 2)), j=0..min(t+1, n/i))))
%p A227135     end:
%p A227135 a:= n-> add(b(n$2, t), t=0..1):
%p A227135 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 15 2017
%t A227135 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1 - t, If[i*(i + 1) < n, 0, Sum[ b[n - i*j, i - 1, Mod[t + j, 2]], {j, 0, Min[t + 1, n/i]}]]];
%t A227135 a[n_] := Sum[b[n, n, t], {t, 0, 1}];
%t A227135 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 21 2018, after _Alois P. Heinz_ *)
%o A227135 (PARI) {A002620(n)=floor(n/2)*ceil(n/2)}
%o A227135 {a(n)=polcoeff(1/(1-x+x*O(x^n)) + sum(m=2,sqrtint(4*n), x^(A002620(m+2)-1)/prod(k=1,m,1-x^k+x*O(x^n))),n)}
%o A227135 for(n=0,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Jul 06 2013
%Y A227135 Cf. A227134 (parts may repeat after odd index).
%K A227135 nonn
%O A227135 0,4
%A A227135 _Joerg Arndt_, Jul 02 2013