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 A122070 Triangle T(n,k), 0<=k<=n, given by T(n,k)=Fibonacci(n+k+1)*binomial(n,k). 3
 1, 1, 2, 2, 6, 5, 3, 15, 24, 13, 5, 32, 78, 84, 34, 8, 65, 210, 340, 275, 89, 13, 126, 510, 1100, 1335, 864, 233, 21, 238, 1155, 3115, 5040, 4893, 2639, 610, 34, 440, 2492, 8064, 16310, 21112, 17080, 7896, 1597, 55, 801, 5184, 19572, 47502, 78860, 82908, 57492, 23256, 4181 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Subtriangle of (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. Mirror image of the triangle in A185384. LINKS FORMULA T(n,k) = A000045(n+k+1)*A007318(n,k) . T(n,n) = Fibonacci(2*n+1) = A001519(n+1) . Sum_{k, 0<=k<=n} T(n,k) = Fibonacci(3*n+1) = A033887(n) . Sum_{k, 0<=k<=n}(-1)^k*T(n,k) =(-1)^n = A033999(n) . Sum_{k, 0<=k<=[n/2]}T(n-k,k)=(Fibonacci(n+1))^2 = A007598(n+1). Sum_{k, 0<=k<=n} T(n,k)*2^k = Fibonacci(4*n+1) = A033889(n). Sum_{k, 0<=k<=n} T(n,k)^2 = A208588(n). G.f.: (1-y*x)/(1-(1+3y)*x-(1+y-y^2)*x^2). T(n,k) = T(n-1,k) + 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. T(n,k) = A185384(n,n-k). T(2n,n) = binomial(2n,n)*Fibonacci(3*n+1) = A208473(n). EXAMPLE Triangle begins: 1; 1, 2; 2, 6, 5; 3, 15, 24, 13; 5, 32, 78, 84, 34; 8, 65, 210, 340, 275, 89; 13, 126, 510, 1100, 1335, 864, 233; (0, 1, 1, -1, 0, 0, ...) DELTA (1, 1, 1, 0, 0, ...) begins : 1 0, 1 0, 1, 2 0, 2, 6, 5 0, 3, 15, 24, 13 0, 5, 32, 78, 84, 34 0, 8, 65, 210, 340, 275, 89 0, 13, 126, 510, 1100, 1335, 864, 233 CROSSREFS Cf. A000045, A001519, A033887, A033889, A185384 Sequence in context: A019749 A209773 A209767 * A181661 A144160 A275142 Adjacent sequences:  A122067 A122068 A122069 * A122071 A122072 A122073 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Oct 15 2006, Mar 13 2012 EXTENSIONS Corrected and extended by Philippe Deléham, Mar 13 2012 STATUS approved

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Last modified April 25 12:19 EDT 2019. Contains 322456 sequences. (Running on oeis4.)