%I #12 Nov 27 2023 17:29:23
%S 1,2,4,7,11,17,25,35,48,64,85,110,141,178,222,275,337,409,493,589,702,
%T 830,977,1144,1333,1549,1792,2065,2372,2714,3100,3528,4005,4534,5119,
%U 5769,6485,7273,8140,9089,10135,11276,12524,13885,15366,16983,18738
%N Number of (unordered) ways of making change for n cents using coins of 1/2, 1, 2, 3, 5, 10, 20, 25, 50, 100 cents (all historical U.S.A. coinage denominations up to 100 cents).
%C The U.S.A. issued the following unusual denomination coins during the 18th and 19th centuries: 1/2-cent pieces, 1793-1857; 2-cent pieces, 1864-1873; 3-cent pieces, 1851-1889; and 20-cent pieces, 1875-1878. This sequence is also the number of ways of making change for n cents using coins of 1 (two types, say, old pre-1858 "large cents" and 1856-to-present "small cents"), 2, 3, 5, 10, 20, 25, 50, 100 cents. For present purposes, one of the two types of 1-cent piece is actually taken to be two 1/2-cent pieces.
%D R. S. Yeoman, A Guide Book of United States Coins, Ed. Kenneth Bressett, 53rd Edition (2000). New York: St. Martin's Press, 1999. pp. 72-77, 92-93, 104-106, 135. (also known as The Official Red Book of United States Coins)
%H Ron Guth, <a href="http://www.coinfacts.com/Administrative/home.html">Your Online Reference For U.S. Coins</a>
%H Mitch Hight, <a href="http://www.coin-gallery.com/cguscoinage.htm">United States Coinage History</a>
%H Clint Leland, <a href="http://www.stanford.edu/~clint/q/">U.S. Coins Quantities Minted</a>
%H <a href="/index/Mag#change">Index entries for sequences related to making change.</a>
%H <a href="/index/Rec#order_217">Index entries for linear recurrences with constant coefficients</a>, order 217.
%F G.f.: 1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^25)*(1-x^50)*(1-x^100))
%e a(2)=4 because change can be made for 2 cents in these 4 ways: (1) 4 1/2-cent coins, (2) 2 1/2-cent, 1 1-cent, (3) 2 1-cent, (4) 1 2-cent coin.
%t CoefficientList[ Series[1 / ((1 - x)^2(1 - x^2)(1 - x^3)(1 - x^5)(1 - x^10)(1 - x^20)(1 - x^25)(1 - x^50)(1 - x^100)), {x, 0, 50} ], x]
%o (PARI) a(n)=polcoeff(1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^25)*(1-x^50)*(1-x^100)+x*O(x^n)), n)
%Y Cf. A067996, A067995, A001314 (two kinds of nickels), A028291 (analog for 1/2, 1, 2, 3, 5 only, or 1(two types), 2, 3, 5 only).
%K easy,nonn
%O 0,2
%A _Rick L. Shepherd_, Feb 08 2002