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A227135
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Partitions with parts repeated at most twice and repetition only allowed if first part has an even index (first index = 1).
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3
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1, 1, 1, 2, 2, 4, 4, 6, 8, 10, 12, 17, 20, 25, 31, 39, 47, 58, 69, 85, 102, 123, 145, 175, 207, 246, 290, 343, 401, 473, 551, 646, 751, 875, 1012, 1177, 1358, 1570, 1807, 2083, 2389, 2746, 3140, 3597, 4106, 4690, 5337, 6082, 6907, 7848, 8895, 10085, 11404, 12902, 14561, 16438, 18520, 20864, 23460, 26385, 29619
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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
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EXAMPLE
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G.f.: 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 8*x^8 +...
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))) +...
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:
01: [ 1:1 2:0 2:1 3:0 5:1 ]
02: [ 1:1 2:0 2:1 4:0 4:1 ]
03: [ 1:1 2:0 2:1 8:0 ]
04: [ 1:1 2:0 3:1 7:0 ]
05: [ 1:1 2:0 4:1 6:0 ]
06: [ 1:1 2:0 10:1 ]
07: [ 1:1 3:0 3:1 6:0 ]
08: [ 1:1 3:0 4:1 5:0 ]
09: [ 1:1 3:0 9:1 ]
10: [ 1:1 4:0 8:1 ]
11: [ 1:1 5:0 7:1 ]
12: [ 1:1 6:0 6:1 ]
13: [ 1:1 12:0 ]
14: [ 2:1 3:0 3:1 5:0 ]
15: [ 2:1 3:0 8:1 ]
16: [ 2:1 4:0 7:1 ]
17: [ 2:1 5:0 6:1 ]
18: [ 2:1 11:0 ]
19: [ 3:1 4:0 6:1 ]
20: [ 3:1 5:0 5:1 ]
21: [ 3:1 10:0 ]
22: [ 4:1 9:0 ]
23: [ 5:1 8:0 ]
24: [ 6:1 7:0 ]
25: [13:1 ]
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MAPLE
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# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, 1-t,
`if`(i*(i+1)<n, 0, add(b(n-i*j, i-1,
irem(t+j, 2)), j=0..min(t+1, n/i))))
end:
a:= n-> add(b(n$2, t), t=0..1):
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MATHEMATICA
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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]}]]];
a[n_] := Sum[b[n, n, t], {t, 0, 1}];
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PROG
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(PARI) {A002620(n)=floor(n/2)*ceil(n/2)}
{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)}
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CROSSREFS
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Cf. A227134 (parts may repeat after odd index).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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