A208904 Triangle of coefficients of polynomials v(n,x) jointly generated with A208660; see the Formula section.

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, 115, 18, 1, 15, 231, 819, 1059, 605, 165, 21, 1, 17, 344, 1652, 2984, 2470, 1032, 224, 24, 1, 19, 489, 3060, 7380, 8378, 4974, 1624, 292, 27, 1, 21, 670, 5301, 16488

Offset

1,2

Comments

For a discussion and guide to related arrays, see A208510.

Riordan array ((1+x)/(1-x)^2, x(1+x)/(1-x)^2) (follows from Kruchinin formula). - Ralf Stephan, Jan 02 2014

From Peter Bala, Jul 21 2014: (Start)

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

/I_k 0\

\ 0 M/

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)

Links

Table of n, a(n) for n=1..59.

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),

v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.

From Vladimir Kruchinin, Mar 11 2013: (Start)

T(n,k) = sum(i=0..n, binomial(i+k-1,2*k-1)*binomial(k,n-i))

((x+x^2)/(1-x)^2)^k = sum(n>=k, T(n,k)*x^n).

T(n,2)=A005900(n).

T(2*n-1,n) / n = A003169(n).

T(2*n,n) = A156894(n), n>1.

sum(k=1..n, T(n,k)) = A003946(n).

sum(k=1..n, T(n,k)*(-1)^(n+k)) = A078050(n).

n*sum(k=1..n, T(n,k)/k) = A058481(n). (End)

Recurrence: T(n+1,k+1) = sum {i = 0..n-k} (2*i + 1)*T(n-i,k). - Peter Bala, Jul 21 2014

Example

First five rows:
1
3...1
5...6....1
7...19...9....1
9...44...42...12...1
First five polynomials v(n,x):
1
3 + x
5 + 6x + x^2
7 + 19x + 9x^2 + x^3
9 + 44x + 42x^2 + 12x^3 + x^4
From Peter Bala, Jul 21 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
/1        \/1        \/1        \      /1            \
|3 1      ||0 1      ||0 1      |      |3  1         |
|5 3 1    ||0 3 1    ||0 0 1    |... = |5  6  1      |
|7 5 3 1  ||0 5 3 1  ||0 0 3 1  |      |7 19  9  1   |
|9 7 5 3 1||0 7 5 3 1||0 0 5 3 1|      |9 44 42 12 1 |
|...      ||...      ||...      |      |...
(End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208660 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208904 *)

Crossrefs

Cf. A208660, A208510. A099375.

Sequence in context: A210551 A113445 A108283 * A344479 A209754 A140950

Adjacent sequences: A208901 A208902 A208903 * A208905 A208906 A208907

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 03 2012

Status

approved