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A025810 Expansion of 1/((1-x^2)*(1-x^5)*(1-x^10)).

%I A025810 #28 Aug 28 2025 11:13:45
%S A025810 1,0,1,0,1,1,1,1,1,1,3,1,3,1,3,3,3,3,3,3,6,3,6,3,6,6,6,6,6,6,10,6,10,
%T A025810 6,10,10,10,10,10,10,15,10,15,10,15,15,15,15,15,15,21,15,21,15,21,21,
%U A025810 21,21,21,21,28,21,28,21,28,28,28,28,28,28,36,28,36,28,36,36,36,36,36,36
%N A025810 Expansion of 1/((1-x^2)*(1-x^5)*(1-x^10)).
%C A025810 Number of partitions of n into parts of size 2, 5, and 10.
%C A025810 a(n) is always a triangular number.
%H A025810 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,0,1,0,-1,0,0,1,0,-1,0,0,-1,0,1).
%F A025810 G.f.: 1/((1-x^2)(1-x^5)(1-x^10)).
%F A025810 Euler transform of length 10 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 1]. - _Michael Somos_, Mar 18 2012
%F A025810 a(n) = a(-17 - n) = a(n - 10) + A008616(n) for all n in Z. - _Michael Somos_, Mar 18 2012
%F A025810 a(n) = A000217( A008616(n) ) = A000008(n) - A000008(n - 1). - _Michael Somos_, Dec 15 2002
%e A025810 G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 3*x^10 + x^11 + 3*x^12 + ...
%t A025810 CoefficientList[Series[1/((1-x^2)(1-x^5)(1-x^10)), {x,0,85}], x]  (* _Harvey P. Dale_, Apr 06 2011 *)
%t A025810 a[ n_] := Module[ {m = Mod[n, 10], k}, k = n - m; If[ m == 1 || m == 3, k -= 10]; k (k + 30) / 200 + 1]; (* _Michael Somos_, Aug 16 2016 *)
%o A025810 (PARI) {a(n) = if( n<-16, a(-17 - n), polcoeff( 1 / ((1 - x^2) * (1 - x^5) * (1 - x^10)) + x * O(x^n), n))}; \\ _Michael Somos_, Mar 18 2012
%o A025810 (PARI) {a(n) = my(m = n%10); n -= m; if( m==1 || m==3, n -= 10); n * (n + 30) / 200 + 1}; \\ _Michael Somos_, Aug 16 2016
%o A025810 (PARI) a(n) = (n^2 + 17*n + (5*n+22)*(-1)^n + 200 + 4*n*[2,-1,1,-2,0][n%5+1])\200 \\ _Hoang Xuan Thanh_, Aug 28 2025
%Y A025810 Cf. A000008, A000217, A008616.
%K A025810 nonn,easy,changed
%O A025810 0,11
%A A025810 _N. J. A. Sloane_