|
%I #21 Sep 08 2022 08:45:01
%S 1,3,12,54,255,1239,6132,30744,155628,793650,4071210,20984340,
%T 108590118,563816526,2935798680,15324533448,80164934919,420151515255,
%U 2205762626010,11597513662350,61060181223195,321870918101535
%N T(2n+1,n), where T is the array in A055830.
%H G. C. Greubel, <a href="/A055835/b055835.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = 3*A055834(n) for n>=1. - _Philippe Deléham_, Jan 25 2014
%F a(n) = Sum_{k=0..n} Sum_{i=k..n} binomial(i,k)*binomial(i+1,n-i)*binomial(n,k). - _Vladimir Kruchinin_, Mar 01 2014
%p seq( `if`(n=0, 1, 3*add(binomial(n+k-1, n)*binomial(k, n-k), k=0..n)), n=0..30); # _G. C. Greubel_, Jan 21 2020
%t Table[If[n==0, 1, 3*Sum[Binomial[k, n-k]*Binomial[n+k-1, n], {k,0,n}]], {n,0,30}] (* _G. C. Greubel_, Jan 21 2020 *)
%o (Maxima) a(n):=sum((sum(binomial(i,k)*binomial(i+1,n-i),i,k,n))*binomial(n,k), k,0,n); /* _Vladimir Kruchinin_, Mar 01 2014 */
%o (PARI) a(n) = if(n==0, 1, 3*sum(k=0, n, binomial(n+k-1, n)*binomial(k, n-k)) ); \\ _Joerg Arndt_, Mar 01 2014
%o (Magma) [1] cat [3*(&+[Binomial(n+k-1, n)*Binomial(k, n-k): k in [0..n]]): n in [1..30]]; // _G. C. Greubel_, Jan 21 2020
%o (Sage) [1]+[3*sum(binomial(n+k-1,n)*binomial(k,n-k) for k in (0..n)) for n in (1..30)] # _G. C. Greubel_, Jan 21 2020
%o (GAP) Concatenation([1], List([1..30], n-> 3*Sum([0..n], k-> Binomial(n+k-1, n) *Binomial(k, n-k) ))); # _G. C. Greubel_, Jan 21 2020
%Y Cf. A055830, A055834.
%K nonn
%O 0,2
%A _Clark Kimberling_, May 28 2000
|
