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A188513 Riordan matrix (1/(x+sqrt(1-4x)),(1-sqrt(1-4x))/(2(x+sqrt(1-4x))). 1
1, 1, 1, 3, 3, 1, 9, 11, 5, 1, 29, 40, 23, 7, 1, 97, 147, 99, 39, 9, 1, 333, 544, 413, 194, 59, 11, 1, 1165, 2025, 1691, 907, 333, 83, 13, 1, 4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1, 14845, 28455, 27464, 17856, 8453, 2979, 775, 143, 17, 1, 53791, 107277, 109631, 76718, 39851, 15804, 4797, 1094, 179, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Triangle begins:

1

1, 1

3, 3, 1

9, 11, 5, 1

29, 40, 23, 7, 1

97, 147, 99, 39, 9, 1

333, 544, 413, 194, 59, 11, 1

1165, 2025, 1691, 907, 333, 83, 13, 1

4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1

First column = sequence A081696

Row sums = sequence A101850

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(n,k) = [x^n] ((1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x)))^k/(x+sqrt(1-4*x)).

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

T(n,k) = sum(binomial(i+k,k)*binomial(2*n-i,n+k+i)*(2*k+3*i+1)/(n+k+i+1), i=0..floor((n-k)/2)).

MATHEMATICA

Flatten[Table[Sum[Binomial[i+k, k]Binomial[2n-i, n+k+i](2k+3i+1)/(n+k+i+1), {i, 0, Floor[(n-k)/2]}], {n, 0, 10}, {k, 0, n}]]

PROG

(Maxima) create_list(sum(binomial(i+k, k)*binomial(2*n-i, n+k+i)*(2*k+3*i+1)/(n+k+i+1), i, 0, floor((n-k)/2)), n, 0, 10, k, 0, n);

CROSSREFS

Cf. A081696, A101850.

Sequence in context: A215120 A084145 A122919 * A216916 A157401 A143911

Adjacent sequences:  A188510 A188511 A188512 * A188514 A188515 A188516

KEYWORD

nonn,easy,tabl

AUTHOR

Emanuele Munarini, Apr 02 2011

STATUS

approved

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Last modified October 1 02:09 EDT 2014. Contains 247498 sequences.