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A208904 Triangle of coefficients of polynomials v(n,x) jointly generated with A208660; see the Formula section. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A209754 A140950 A256504

Adjacent sequences:  A208901 A208902 A208903 * A208905 A208906 A208907

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 03 2012

STATUS

approved

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Last modified October 15 23:54 EDT 2019. Contains 328038 sequences. (Running on oeis4.)