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A190252 Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2*(1+x)). 4
1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 12, 14, 9, 4, 1, 31, 38, 28, 14, 5, 1, 83, 106, 84, 48, 20, 6, 1, 227, 301, 252, 157, 75, 27, 7, 1, 634, 864, 758, 504, 265, 110, 35, 8, 1, 1799, 2508, 2283, 1602, 906, 417, 154, 44, 9, 1, 5171, 7348, 6897, 5056, 3035, 1512, 623, 208, 54, 10, 1, 15027, 21699, 20903, 15894, 10020, 5324, 2387, 894, 273, 65, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
First column = A071359(n+1).
Central coefficients = A190253.
Row sums = A190254.
Diagonal sums = A190255.
LINKS
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
FORMULA
T(n,k) = [x^(n-k)]g(x)^(k+1), where g(x) = (1-x-sqrt(1-2*x-3*x^2-4*x^3)) / (2*x^2*(1+x)).
T(n,k) = sum(i=0..(n-k)/2, binomial(2*i+k,i)*(k+1)/(i+k+1) * sum(j=0..i, binomial(i,j)*binomial(n-j,2*i+k) ) ).
Recurrence: T(n+2,k+1) = T(n+1,k) + T(n+1,k+1) + T(n+1,k+2) + T(n,k+2).
EXAMPLE
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 3, 1;
12, 14, 9, 4, 1;
31, 38, 28, 14, 5, 1;
83, 106, 84, 48, 20, 6, 1;
227, 301, 252, 157, 75, 27, 7, 1;
634, 864, 758, 504, 265, 110, 35, 8, 1;
...
MATHEMATICA
Flatten[Table[Sum[Binomial[2i+k, i](k+1)/(i+k+1) Sum[Binomial[i, j] Binomial[n-j, 2i+k], {j, 0, i}], {i, 0, (n-k)/2}], {n, 0, 12}, {k, 0, n}]]
PROG
(Maxima) create_list(sum(binomial(2*i+k, i)*(k+1)/(i+k+1)*sum( binomial(i, j)*binomial(n-j, 2*i+k), j, 0, i), i, 0, (n-k)/2), n, 0, 12, k, 0, n);
CROSSREFS
Sequence in context: A178518 A299499 A190215 * A141751 A079222 A033184
KEYWORD
nonn,tabl
AUTHOR
Emanuele Munarini, May 06 2011
STATUS
approved

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Last modified February 21 04:08 EST 2024. Contains 370219 sequences. (Running on oeis4.)