cp's OEIS Frontend

A052156 Number of compositions of n into 2*j-1 kinds of j's for all j>=1.

%I A052156 #33 Jun 29 2023 11:49:44
%S A052156 1,1,4,12,36,108,324,972,2916,8748,26244,78732,236196,708588,2125764,
%T A052156 6377292,19131876,57395628,172186884,516560652,1549681956,4649045868,
%U A052156 13947137604,41841412812,125524238436,376572715308,1129718145924
%N A052156 Number of compositions of n into 2*j-1 kinds of j's for all j>=1.
%C A052156 First differences of A025192, also second differences of A000244.
%D A052156 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
%D A052156 P. Ribenhoim, The Little Book of Big Primes, Springer-Verlag, N.Y., 1991, p. 53.
%H A052156 Vincenzo Librandi, <a href="/A052156/b052156.txt">Table of n, a(n) for n = 0..1000</a>
%H A052156 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (3).
%F A052156 a(n) = 4*3^(n-2); n >= 2; a(0) = 1; a(1) = 1.
%F A052156 G.f.: (1-x)^2/(1-3*x).
%F A052156 G.f.: 1/(1-sum(j>=1, (2*j-1)*x^j )). - _Joerg Arndt_, Jul 06 2011
%F A052156 a(n) = 3*a(n-1)+(-1)^n*C(2, 2-n).
%F A052156 a(n) = A003946(n-1), n>0. - _R. J. Mathar_, Oct 13 2008
%F A052156 a(n) = (-4*n + 9) * a(n-1) + 3 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - _Michael Somos_, Jul 23 2011
%F A052156 a(n) = Sum_{k, 0<=k<=n} A201780(n,k). - _Philippe Deléham_, Dec 05 2011
%e A052156 1 + x + 4*x^2 + 12*x^3 + 36*x^4 + 108*x^5 + 324*x^6 + 972*x^7 + 2916*x^8 + ...
%t A052156 CoefficientList[Series[(1 - x)^2/(1 - 3 x), {x, 0, 40}], x ] (* _Vincenzo Librandi_, Apr 29 2014 *)
%o A052156 (PARI) {a(n) = local(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-4*k + 9) * A[k-1] + 3 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* _Michael Somos_, Jul 23 2011 */
%Y A052156 Cf. A025192, A000244, A003462.
%K A052156 easy,nonn
%O A052156 0,3
%A A052156 _Barry E. Williams_, Jan 24 2000
%E A052156 New name from _Joerg Arndt_, Jul 06 2011