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A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

%I A126348 #30 Nov 12 2024 19:22:21
%S A126348 1,1,2,4,7,12,20,33,53,84,131,202,308,465,695,1030,1514,2209,3201,
%T A126348 4609,6596,9386,13284,18705,26211,36561,50776,70226,96742,132765,
%U A126348 181540,247369,335940,454756,613689,825698,1107755,1482038,1977465,2631664
%N A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).
%C A126348 In triangle A126347, row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).
%C A126348 Row sums of A253830. a(n) equals the number of colored compositions of n, as defined in A253830,  whose associated color partition has distinct parts. An example is given below. - _Peter Bala_, Jan 20 2015
%H A126348 Vaclav Kotesovec, <a href="/A126348/b126348.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..500 from Seiichi Manyama)
%F A126348 1 + Sum_{k>0} x^(k * (k + 1) / 2) / ((1 - x)^k * (1 - x) * (1 - x^2) ... (1 - x^k)). - _Michael Somos_, Aug 17 2008
%F A126348 G.f.: Product_{k>0} (1+x^k/(1-x)). - _Vladeta Jovovic_, Oct 05 2008
%F A126348 G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)/(d*(1 - x)^d)). - _Ilya Gutkovskiy_, Apr 19 2019
%e A126348 a(5) = 12: The colored compositions (defined in A253830) of 5 whose color partitions have distinct parts are
%e A126348 5(c1), 5(c2), 5(c3), 5(c4), 5(c5),
%e A126348 1(c1) + 4(c2), 1(c1) + 4(c3), 1(c1) + 4(c4),
%e A126348 3(c1) + 2(c2),
%e A126348 2(c1) + 3(c2), 2(c1) + 3(c3), 2(c2) + 3(c3). - _Peter Bala_, Jan 20 2015
%t A126348 nmax = 50; CoefficientList[Series[Product[(1 - x + x^k)/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 16 2019 *)
%o A126348 (PARI) {B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
%o A126348 {a(n)=Vec(B(n+1,'q)+O('q^(n*(n-1)/2+1)))[n*(n-1)/2+1]}
%o A126348 (PARI) {a(n) = local(t); if( n<0, 0, t = 1; polcoeff( sum(k=1, (sqrtint(8*n + 1) - 1)\2, t = t * x^k / (1 - x) / (1 - x^k) + x * O(x^n), 1), n))} /* _Michael Somos_, Aug 17 2008 */
%Y A126348 Cf. A126347, A126349; factorial variant: A126471. A253830, A307599, A307601, A307602.
%K A126348 nonn
%O A126348 0,3
%A A126348 _Paul D. Hanna_, Dec 31 2006