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 A000710 Number of partitions of n, with two kinds of 1, 2, 3 and 4. (Formerly M1375 N0535) 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also number of partitions of 2*n+4 with exactly 4 odd parts. - Vladeta Jovovic, Jan 12 2005 Convolution of A000041 and A001400. - Vaclav Kotesovec, Aug 18 2015 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 REFERENCES 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 N. J. A. Sloane, Transforms FORMULA 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 EXAMPLE a(2) = 5 because we have 2, 2', 1+1, 1+1', 1'+1'. MAPLE 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 MATHEMATICA 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 *) CROSSREFS Cf. A000712. Cf. A000070, A008951, A000097, A000098. Fifth column of Riordan triangle A008951 and of triangle A103923. Sequence in context: A263002 A325649 A325719 * A160461 A117487 A263348 Adjacent sequences:  A000707 A000708 A000709 * A000711 A000712 A000713 KEYWORD nonn,easy AUTHOR EXTENSIONS Edited by Emeric Deutsch, Mar 22 2005 STATUS approved

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Last modified June 1 12:49 EDT 2020. Contains 334762 sequences. (Running on oeis4.)