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Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
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%I #33 Oct 19 2022 10:58:03

%S 1,0,1,0,2,1,0,3,4,1,0,5,10,6,1,0,8,22,21,8,1,0,13,45,59,36,10,1,0,21,

%T 88,147,124,55,12,1,0,34,167,339,366,225,78,14,1,0,55,310,741,976,770,

%U 370,105,16,1,0,89,566,1557,2422,2337,1443,567,136,18,1,0,144,1020,3174,5696,6505,4920,2485,824,171,20,1

%N Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

%C A Fibonacci convolution triangle; Riordan array (1, x*(1+x)/(1-x-x^2)).

%H Alois P. Heinz, <a href="/A155112/b155112.txt">Rows n = 0..140, flattened</a>

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

%F Explicit formula: T(n,k) = Sum_{i=0..floor(n/2)} binomial(n-i, i)*binomial(n-i, k)*k/(n-i), for n > 0.

%F G.f.: (1-x-x^2)/(1-(1+y)*x-(1+y)*x^2). - _Philippe Deléham_, Feb 21 2012

%F Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A155020(n), A154964(n), A154968(n), A154996(n), A154997(n), A154999(n), A155000(n), A155001(n), A155017(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.

%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - _Philippe Deléham_, Feb 21 2012

%F Sum_{k=0..n} T(n, k)*(m-1)^k = (1/m)*[n=0] - (m-1)*(i*sqrt(m))^(n-2)*ChebyshevU(n, -i*sqrt(m)/2). - _G. C. Greubel_, Mar 26 2021

%F Sum_{k=0..n} k * T(n,k) = A291385(n-1) for n>=1. - _Alois P. Heinz_, Sep 29 2022

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 0, 3, 4, 1;

%e 0, 5, 10, 6, 1;

%e 0, 8, 22, 21, 8, 1;

%e 0, 13, 45, 59, 36, 10, 1;

%e ...

%p # Uses function PMatrix from A357368.

%p PMatrix(10, n -> combinat:-fibonacci(n+1)); # _Peter Luschny_, Oct 19 2022

%t T[n_, k_]:= If[n==0, 1, Sum[Binomial[n-j, j]*Binomial[n-j, k]*k/(n-j), {j, 0, Floor[n/2]}]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 26 2021 *)

%o (Magma)

%o T:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n-j,j)*Binomial(n-j,k)*k/(n-j): j in [0..Floor(n/2)]]) >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 26 2021

%o (Sage)

%o def T(n,k): return 1 if n==0 else sum( binomial(n-j,j)*binomial(n-j,k)*k/(n-j) for j in (0..n//2) )

%o flatten([[T(n,k) for k in [0..n]] for n in [0..12]]) # _G. C. Greubel_, Mar 26 2021

%Y Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.

%Y Cf. A000012, A155020, A154964, A154968, A154996, A154997, A154999, A155000, A155001, A155017.

%Y Cf. A000045, A154929.

%Y T(2n,n) gives A262910.

%Y Cf. A291385.

%K nonn,tabl

%O 0,5

%A _Philippe Deléham_, Jan 20 2009

%E Typos in two terms corrected by _Alois P. Heinz_, Aug 08 2015