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%I #46 Jun 05 2021 22:07:35
%S 1,1,2,2,3,4,5,6,7,8,11,12,15,16,19,22,25,28,31,34,40,43,49,52,58,65,
%T 71,78,84,91,102,109,120,127,138,151,162,175,186,199,217,230,248,261,
%U 279,300,318,339,357,378,407,428,457,478,507,540,569,602,631,664
%N Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25, 50 cents.
%C Number of partitions of n into parts 1, 2, 5, 10, 25, and 50. - _Joerg Arndt_, Sep 05 2014
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H T. D. Noe, <a href="/A001302/b001302.txt">Table of n, a(n) for n = 0..1000</a>
%H H. Bottomley, <a href="/A000008/a000008.gif">Initial terms of A000008, A001301, A001302, A001312, A001313</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=178">Encyclopedia of Combinatorial Structures 178</a>
%H <a href="/index/Mag#change">Index entries for sequences related to making change.</a>
%H <a href="/index/Rec#order_93">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 1, -1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1).
%F G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^25)*(1-x^50)).
%F a(n) = Sum_{k=0..floor(n/2)} A001300(n-2*k). - _Christian Krause_, Apr 24 2021
%t CoefficientList[ Series[ 1 / ((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)(1 - x^25)(1 - x^50)), {x, 0, 55} ], x ]
%t Array[Length@IntegerPartitions[#, All, {1, 2, 5, 10, 25, 50}]&, 100, 0] (* _Giorgos Kalogeropoulos_, Apr 24 2021 *)
%o (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^25)*(1-x^50))+ O(x^100)) \\ _Michel Marcus_, Sep 05 2014
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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