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A000710
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Number of partitions of n, with two kinds of 1, 2, 3 and 4.
(Formerly M1375 N0535)
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12
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1, 2, 5, 10, 20, 35, 62, 102, 167, 262, 407, 614, 919, 1345, 1952, 2788, 3950, 5524, 7671, 10540, 14388, 19470, 26190, 34968, 46439, 61275, 80455, 105047, 136541, 176593, 227460, 291673, 372605, 474085, 601105, 759380, 956249, 1200143, 1501749, 1873407
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OFFSET
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0,2
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COMMENTS
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Also number of partitions of 2*n+4 with exactly 4 odd parts. - Vladeta Jovovic, Jan 12 2005
Also the sum of binomial (D(p), 4) over partitions p of n+10, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018
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REFERENCES
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H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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).
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LINKS
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FORMULA
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Euler transform of 2 2 2 2 1 1 1...
G.f.: 1/((1-x)(1-x^2)(1-x^3)(1-x^4)*Product_{k>=1} (1-x^k)).
a(n) = Sum_{j=0..floor(n/4)} A000098(n-4*j), n >= 0.
a(n) ~ sqrt(3)*n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - Vaclav Kotesovec, Aug 18 2015
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EXAMPLE
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a(2) = 5 because we have 2, 2', 1+1, 1+1', 1'+1'.
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MAPLE
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with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<5, 2, 1)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
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MATHEMATICA
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etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[If[#<5, 2, 1]&]; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^4))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@4], {n, 0, 39}] (* Robert Price, Jul 28 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[_, _] = 0;
a[n_] := T[n + 10, 4];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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