A114422 - OEIS

%I #11 Sep 08 2022 08:45:23

%S 1,1,1,3,3,1,7,9,5,1,19,26,19,7,1,51,75,65,33,9,1,141,216,211,132,51,

%T 11,1,393,623,665,483,235,73,13,1,1107,1800,2058,1674,963,382,99,15,1,

%U 3139,5211,6294,5598,3663,1739,581,129,17,1,8953,15115,19095,18261,13243

%N Riordan array (1/sqrt(1-2*x-3*x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.

%C First column is central trinomial numbers A002426.

%C Second column is A005774.

%C Third column is A025568.

%C Row sums are A116387.

%C Diagonal sums are A116388.

%C Product of A007318 and A116382.

%C Column k has e.g.f. exp(x)*Sum_{j=0..k} C(k,j)*Bessel_I(k+j,2*x).

%H G. C. Greubel, <a href="/A114422/b114422.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F Riordan array (1/sqrt(1-2*x-3*x^2), (1-x-2*x^2-sqrt(1-2*x-3*x^2) ) / (2*x^2)).

%F Number triangle T(n,k) = Sum_{j=0..n} C(n,j-k)*C(j,n-j).

%e Triangle begins

%e 1,

%e 1, 1,

%e 3, 3, 1,

%e 7, 9, 5, 1,

%e 19, 26, 19, 7, 1,

%e 51, 75, 65, 33, 9, 1,

%e 141, 216, 211, 132, 51, 11, 1

%t T[n_, k_] := Sum[Binomial[n, j - k]*Binomial[j, n - j], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* _G. C. Greubel_, Feb 28 2017 *)

%o (PARI) {T(n,k) = sum(j=0,n, binomial(n, j-k)*binomial(j, n-j))};

%o for(n=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 15 2018

%o (Magma) [[(&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Dec 15 2018

%o (Sage) [[sum(binomial(n, j-k)*binomial(j, n-j) for j in range(n+1)) for k in range(n+1)] for n in range(10)] # _G. C. Greubel_, Dec 15 2018

%o (GAP) T:=Flat(List([0..10], n->List([0..n], k->Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j))))); # _G. C. Greubel_, Dec 15 2018

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Feb 12 2006