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%I #34 Aug 22 2015 01:57:18
%S 1,3,1,5,6,1,7,19,9,1,9,44,42,12,1,11,85,138,74,15,1,13,146,363,316,
%T 115,18,1,15,231,819,1059,605,165,21,1,17,344,1652,2984,2470,1032,224,
%U 24,1,19,489,3060,7380,8378,4974,1624,292,27,1,21,670,5301,16488
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208660; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%C Riordan array ((1+x)/(1-x)^2, x(1+x)/(1-x)^2) (follows from Kruchinin formula). - _Ralf Stephan_, Jan 02 2014
%C From _Peter Bala_, Jul 21 2014: (Start)
%C Let M denote the lower unit triangular array A099375 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
%C /I_k 0\
%C \ 0 M/
%C having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)
%F u(n,x)=u(n-1,x)+2x*v(n-1,x),
%F v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%F From _Vladimir Kruchinin_, Mar 11 2013: (Start)
%F T(n,k) = sum(i=0..n, binomial(i+k-1,2*k-1)*binomial(k,n-i))
%F ((x+x^2)/(1-x)^2)^k = sum(n>=k, T(n,k)*x^n).
%F T(n,2)=A005900(n).
%F T(2*n-1,n) / n = A003169(n).
%F T(2*n,n) = A156894(n), n>1.
%F sum(k=1..n, T(n,k)) = A003946(n).
%F sum(k=1..n, T(n,k)*(-1)^(n+k)) = A078050(n).
%F n*sum(k=1..n, T(n,k)/k) = A058481(n). (End)
%F Recurrence: T(n+1,k+1) = sum {i = 0..n-k} (2*i + 1)*T(n-i,k). - _Peter Bala_, Jul 21 2014
%e First five rows:
%e 1
%e 3...1
%e 5...6....1
%e 7...19...9....1
%e 9...44...42...12...1
%e First five polynomials v(n,x):
%e 1
%e 3 + x
%e 5 + 6x + x^2
%e 7 + 19x + 9x^2 + x^3
%e 9 + 44x + 42x^2 + 12x^3 + x^4
%e From _Peter Bala_, Jul 21 2014: (Start)
%e With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
%e /1 \/1 \/1 \ /1 \
%e |3 1 ||0 1 ||0 1 | |3 1 |
%e |5 3 1 ||0 3 1 ||0 0 1 |... = |5 6 1 |
%e |7 5 3 1 ||0 5 3 1 ||0 0 3 1 | |7 19 9 1 |
%e |9 7 5 3 1||0 7 5 3 1||0 0 5 3 1| |9 44 42 12 1 |
%e |... ||... ||... | |...
%e (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A208660 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A208904 *)
%Y Cf. A208660, A208510. A099375.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 03 2012
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