A208904 - OEIS

%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