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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
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
Sequence in context: A210551 A113445 A108283 * A344479 A209754 A140950
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 03 2012
STATUS
approved

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Last modified June 17 11:50 EDT 2024. Contains 373445 sequences. (Running on oeis4.)