|
%I #30 Aug 08 2018 15:11:03
%S 1,4,1,24,8,1,176,64,12,1,1440,544,120,16,1,12608,4864,1168,192,20,1,
%T 115584,45184,11424,2112,280,24,1,1095424,432128,113088,22528,3440,
%U 384,28,1,10646016,4227584,1133952,237824,39840,5216,504,32,1,105522176,42115072,11506944,2505728,448064,65280,7504,640,36,1
%N Triangle T(n,k) for A(x)^k=sum(n>=k T(n,k)*x^n), where o.g.f. A(x) satisfies A(x)=(a+b*x*A(x))/(c-d*x*A(x)), a=1,b=2,c=1,d=2.
%C For o.g.f G(x), G(A(x,a,b,c,d))=g(0)+sum(n>0, sum(k=1..n, T(n,k,a,b,c,d)*g(k))x^n).
%C T(n,k,1,1,1,1)=A080247(n,k),
%C T(n,k,2,-1,1,1)=A108891(n,k),
%C T(n,k,1,-2,1,1)=A125692(n,k),
%C T(n,k,1,-3,1,1)=A125694(n,k),
%C T(n,k,-2,1,1,1)=A085403(n,k).
%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F T(n,k,a,b,c,d):=k/n*sum(i=0..n-k, binomial(n,n-k-i)*a^(k+i)*b^(n-k-i)*binomial(i+n-1,n-1)*c^(-i-n)*d^i), a,b,c,d !=0, n>0.
%F T(n,k,1,2,1,2):=k/n*2^(n-k)*sum(i=0..n-k, binomial(n,n-k-i)*binomial(i+n-1,n-1)), n>0.
%F Conjecture: T(n,1) = A156017(n-1). - _R. J. Mathar_, Nov 14 2011
%e 1,
%e 4,1,
%e 24,8,1,
%e 176,64,12,1,
%e 1440,544,120,16,1,
%e 12608,4864,1168,192,20,1,
%e 115584,45184,11424,2112,280,24,1,
%e 1095424,432128,113088,22528,3440,384,28,1,
%e 10646016,4227584,1133952,237824,39840,5216,504,32,1,
%e 105522176,42115072,11506944,2505728,448064,65280,7504,640,36,1
%t T[n_, k_, a_, b_, c_, d_] := k/n Sum[Binomial[n, n - k - i] a^(k + i) b^(n - k - i) Binomial[i + n - 1, n - 1] c^(-i - n) d^i, {i, 0, n - k}];
%t T[n_, k_] := T[n, k, 1, 2, 1, 2];
%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 08 2018, from formula *)
%K nonn,tabl
%O 1,2
%A _Vladimir Kruchinin_, Feb 12 2011
|
