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%I #2 Sep 24 2013 09:23:05
%S 1,1,1,1,1,2,3,3,3,3,5,7,7,7,7,10,15,15,15,15,19,25,25,25,25,31,41,41,
%T 41,41,49,63,63,63,63,74,95,95,95,95,111,147,147,147,147,166,209,209,
%U 209,209,234,293,293,293,293,322,391,391,391,391,427,515,515,515,515
%N Skylar (age 7) counts change by stacking all coins of the same type then arranging the stacks in a row. a(n) is the number of distinct Skylar stackings of n cents using any combination of pennies, nickels, dimes or quarters.
%C Sequence definition and Scratch program to compute the 100 terms due to Skylar Sutherland. Generating function contributed by Andrew V. Sutherland. Related to A001299, but distinguishes permutations of coin types.
%D Skylar Sutherland, student presentation at "The Undiscovered Country", a course for young mathematicians. Part of MIT's Educational Studies Program.
%F Let A_v(x,y) = 1-y+y/(1-x)^v and A(x,y) = A_1(x,y)A_5(x,y)A_10(x,y)A_25(x,y). Let A^(k)(x,y) denote the k-th partial derivative of A(x,y) w.r.t. y. The generating function of a(n) is A(x) = Sum A^(k)(x,0) for k from 0 to 4.
%e a(16) = 15 = 1+2*4+6*1 since the distinct Skylar stackings of 16 cents are:
%e 16p, 11p1n, 1n11p, 6p2n, 2n6p, 1p3n, 3n1p, 1p1d, 1d1p, 1p1n1d, 1p1d1n, 1n1p1d, 1n1d1p, 1d1p1n, 1d1n1p
%Y Cf. A001299.
%K nonn
%O 0,6
%A _Andrew V. Sutherland_, Aug 20 2007
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