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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

%I #20 Jun 06 2023 18:11:03

%S 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,

%T 227,301,252,157,75,27,7,1,634,864,758,504,265,110,35,8,1,1799,2508,

%U 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

%N 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)).

%C First column = A071359(n+1).

%C Central coefficients = A190253.

%C Row sums = A190254.

%C Diagonal sums = A190255.

%H G. C. Greubel, <a href="/A190252/b190252.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

%F 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)).

%F 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) ) ).

%F Recurrence: T(n+2,k+1) = T(n+1,k) + T(n+1,k+1) + T(n+1,k+2) + T(n,k+2).

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 5, 5, 3, 1;

%e 12, 14, 9, 4, 1;

%e 31, 38, 28, 14, 5, 1;

%e 83, 106, 84, 48, 20, 6, 1;

%e 227, 301, 252, 157, 75, 27, 7, 1;

%e 634, 864, 758, 504, 265, 110, 35, 8, 1;

%e ...

%t 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}]]

%o (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);

%Y Cf. A190253, A190254, A190255.

%K nonn,tabl

%O 0,4

%A _Emanuele Munarini_, May 06 2011

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)