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%I M1002 N0375 #49 Jul 16 2023 20:10:10
%S 1,2,4,6,9,13,18,24,31,39,50,62,77,93,112,134,159,187,218,252,292,335,
%T 384,436,494,558,628,704,786,874,972,1076,1190,1310,1440,1580,1730,
%U 1890,2060,2240,2435,2640,2860,3090,3335,3595,3870,4160,4465,4785,5126
%N Partial sums of (unordered) ways of making change for n cents using coins of 1, 2, 5, 10 cents.
%C Number of partitions of n into two kinds of part 1 and one kind of parts 2, 5, and 10. - _Joerg Arndt_, May 10 2014
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Christian G. Bower, <a href="/A000064/b000064.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1,1,-2,0,2,-1,1,-2,0,2,-1,-1,2,0,-2,1).
%F G.f.: 1 / ( ( 1 - x )^2 * ( 1 - x^2 ) * ( 1 - x^5 ) * ( 1 - x^10 ) ).
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-8) - a(n-9) + a(n-10) - 2*a(n-11) + 2*a(n-13) - a(n-14) - a(n-15) + 2*a(n-16) - 2*a(n-18) + a(n-19). - _Fung Lam_, May 07 2014
%F a(n) ~ n^4 / 2400 as n->oo. - _Daniel Checa_, Jul 11 2023
%p 1/(1-x)^2/(1-x^2)/(1-x^5)/(1-x^10)
%p a:= proc(n) local m, r; m := iquo(n, 10, 'r'); r:= r+1; (55+(119+(95+ 25*m) *m) *m) *m/6+ [1, 2, 4, 6, 9, 13, 18, 24, 31, 39][r]+ [0, 26, 61, 99, 146, 202, 267, 341, 424, 516][r]*m/6+ [0, 10, 21, 33, 46, 60, 75, 91, 108, 126][r]*m^2/2+ (5*r-5) *m^3/3 end: seq(a(n), n=0..100); # _Alois P. Heinz_, Oct 05 2008
%t CoefficientList[Series[1/((1-x)^2(1-x^2)(1-x^5)(1-x^10)),{x,0,100}],x] (* _Vladimir Joseph Stephan Orlovsky_, Jan 25 2012 *)
%o (PARI) a(n)=if(n<0,0,polcoeff(1/((1-x)^2*(1-x^2)*(1-x^5)*(1-x^10))+x*O(x^n),n))
%o (PARI) a(n)=floor((n^4+38*n^3+476*n^2+2185*n+3735)/2400+(n+1)*(-1)^n/160+(n\5+1)*[0,0,1,0,-1][n%5+1]/10) \\ _Tani Akinari_, May 10 2014
%Y Cf. A000008.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Corrected and extended by _Simon Plouffe_
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