|
%I #17 Sep 08 2022 08:45:24
%S 1,1,1,3,4,1,7,13,7,1,19,42,32,10,1,51,131,128,60,13,1,141,406,475,
%T 292,97,16,1,393,1247,1685,1267,561,143,19,1,1107,3814,5800,5112,2804,
%U 962,198,22,1,3139,11623,19540,19624,12748,5464,1522,262,25,1,8953,35334
%N Riordan array (1/sqrt(1-2*x-3*x^2), 1/sqrt(1-2*x-3*x^2) -1).
%C Triangle, read by rows, given by [1, 2, -1, -1, 2, 1/2, 1/2, 2, -1, -1, 2, 1/2, 1/2, 2, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 11 2020
%H G. C. Greubel, <a href="/A116392/b116392.txt">Rows n = 0..100 of triangle, flattened</a>
%F Number triangle T(n,k) = Sum_{j=0..n} C(n,j)*A116389(j,k).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 3, 4, 1;
%e 7, 13, 7, 1;
%e 19, 42, 32, 10, 1;
%e 51, 131, 128, 60, 13, 1;
%p # The function RiordanSquare is defined in A321620.
%p RiordanSquare(1/sqrt(1 - 2*x - 3*x^2), 10); # _Peter Luschny_, Feb 15 2020
%t t[n_,k_]:= Sum[(-1)^(k-j)*Binomial[k,j]*Sum[4^r*Binomial[r+(j-1)/2, r]* Binomial[j, n-2*r], {r,0,Floor[n/2]}], {j,0,k}]; Table[Sum[Binomial[n, j]*t[j,k], {j,0,n}] {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 23 2019 *)
%o (PARI) t(n,k) = sum(j=0,k, sum(r=0,floor(n/2), (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) ));
%o T(n,k) = sum(j=0,n, binomial(n,j)*t(j,k)); \\ _G. C. Greubel_, May 23 2019
%o (Magma) [[(&+[ Binomial(n,m)*(&+[ (&+[ Round((-1)^(k-j)*4^r* Binomial(k,j)*Binomial(j, m-2*r)*Gamma(r+(j+1)/2)/(Factorial(r)*Gamma((j+1)/2))) : r in [0..Floor(n/2)]]) : j in [0..k]]): m in [0..n]]) : k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 23 2019
%o (Sage) [[sum(binomial(n,m)*sum( sum( (-1)^(k-j)*4^r* binomial(k,j)* binomial(r+(j-1)/2, r)*binomial(j, m-2*r) for r in (0..floor(n/2))) for j in (0..k)) for m in (0..n)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 23 2019
%Y Row sums are A115967. Diagonal sums are A116394.
%Y Cf. A321620.
%K easy,nonn,tabl
%O 0,4
%A _Paul Barry_, Feb 12 2006
|
