A007209 - OEIS

%I M0315 #39 May 07 2017 22:02:36

%S 1,1,2,2,4,5,7,9,12,16,20,25,32,39,49,58,73,86,105,123,149,175,207,

%T 241,284,331,385,444,515,592,682,777,894,1015,1160,1310,1492,1683,

%U 1903,2140,2412,2708,3037,3395,3801,4239,4730,5254,5852,6489,7204,7965,8823,9741

%N Number of partitions of n into parts of sizes {a( )} is a(n).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A007209/b007209.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%F G.f. A(x) = 1 / ((1 - x) * Product_{k>2} (1 - x^a(k))).

%e G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 12*x^8 + 16*x^9 + ...

%p a:= proc(n) option remember; local x, j, i; `if`(n<6, [1, 1, 2, 2, 4, 5][n+1], coeff(series(1 /mul(1-x^i, i=[{seq(a(j), j=1..n-1)}[]]), x, n+1), x, n)) end: seq(a(n), n=0..60); # _Alois P. Heinz_, Apr 27 2009

%t a[0] = a[1] = 1; a[2] = a[3] = 2; a[4] = 4; a[5] = 5; a[n_] := a[n] = SeriesCoefficient[ Series[ 1/((1 - x)*Product[ 1 - x^a[k], {k, 3, n-1}]), {x, 0, n}], n]; Table[a[n], {n, 0, 53}] (* _Jean-François Alcover_, Dec 09 2011 *)

%o (PARI) {a(n) = local(A, k); A = Ser([ 1, 1, 2, 2, 4, 5]); while( n > (k = #A - 1), A = 1 / (1 - x) / prod( i=3, k, 1 - x^polcoeff(A, i), 1 + x^2 * O(x^ polcoeff( A, k))) ); polcoeff( A, n)}; /* _Michael Somos_, Aug 08 2011 */

%K nonn,nice

%O 0,3

%A _N. J. A. Sloane_, _Mira Bernstein_

%E More terms from _Alois P. Heinz_, Apr 27 2009