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

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, 6, 10, 10, 10, 10, 10, 10, 15, 10, 15, 10, 15, 15, 15, 15, 15, 15, 21, 15, 21, 15, 21, 21, 21, 21, 21, 21, 28, 21, 28, 21, 28, 28, 28, 28, 28, 28, 36, 28, 36, 28, 36, 36, 36, 36, 36, 36

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

Number of partitions of n into parts of size 2, 5, and 10.

a(n) is always a triangular number.

Examples

			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 + ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,1,0,-1,0,0,1,0,-1,0,0,-1,0,1).

Crossrefs

Cf. A000008, A000217, A008616.

Programs

Mathematica

CoefficientList[Series[1/((1-x^2)(1-x^5)(1-x^10)), {x,0,85}], x]  (* Harvey P. Dale, Apr 06 2011 *)
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 *)

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

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

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

Formula

G.f.: 1/((1-x^2)(1-x^5)(1-x^10)).

Euler transform of length 10 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 1]. - Michael Somos, Mar 18 2012

a(n) = a(-17 - n) = a(n - 10) + A008616(n) for all n in Z. - Michael Somos, Mar 18 2012

a(n) = A000217( A008616(n) ) = A000008(n) - A000008(n - 1). - Michael Somos, Dec 15 2002