A201166 - OEIS

%I #16 Apr 02 2020 07:48:26

%S 1,2,1,5,4,1,12,14,6,1,31,46,27,8,1,85,150,108,44,10,1,248,493,410,

%T 206,65,12,1,762,1644,1519,887,348,90,14,1,2440,5569,5569,3641,1673,

%U 542,119,16,1,8064,19147,20348,14524,7529,2876,796,152,18,1,27300,66706,74367,56925,32458,14077,4620,1118,189,20,1

%N Triangle read by rows: the Fibonacci triangle times Pascal's triangle (A007318).

%H Tian-Xiao He and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2008.11.021">Sequence characterization of Riordan arrays</a>, Discrete Math. 309 (2009), no. 12, 3962-3974.

%e Triangle begins:

%e 1

%e 2 1

%e 5 4 1

%e 12 14 6 1

%e 31 46 27 8 1

%e 85 150 108 44 10 1

%e 248 493 410 206 65 12 1

%e ...

%p A201166 := proc(n,k)

%p     add( A139375(n,j)*binomial(j,k),j=k..n) ;

%p end proc: # _R. J. Mathar_, Jul 09 2013

%t F[n_, k_] := If[k == 0, Fibonacci[n+1], k Sum[Fibonacci[i+1] Binomial[2(n-i)-k-1, n-i-1]/(n-i), {i, 0, n-k}]];

%t T[n_, k_] := Sum[F[n, j] Binomial[j, k], {j, k, n}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 02 2020 *)

%Y Cf. A007318, A139375, A201165.

%K nonn,tabl

%O 0,2

%A _N. J. A. Sloane_, Nov 27 2011