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 A178534 Triangle T(n,k) read by rows. T(n,1)=A000045(n+1), k>1: T(n,k) = (sum from i = 1 to k-1 of T(n-i,k-1)) - (sum from i = 1 to k-1 of T(n-i,k)). 2
 1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 8, 3, 1, 1, 1, 13, 5, 3, 1, 1, 1, 21, 8, 4, 2, 1, 1, 1, 34, 13, 6, 4, 2, 1, 1, 1, 55, 21, 11, 6, 3, 2, 1, 1, 1, 89, 34, 17, 9, 6, 3, 2, 1, 1, 1, 144, 55, 27, 15, 9, 5, 3, 2, 1, 1, 1, 233, 89, 45, 25, 14, 9, 5, 3, 2, 1, 1, 1, 377, 144, 72, 40, 23, 14, 8, 5, 3, 2, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA T(n,1) = A000045(n+1), k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) - Sum_{i=1..k-1} T(n-i,k). T(n,k) = A129713*A051731. [Mats Granvik, Oct 22 2010] Comment from R. J. Mathar, Sep 16 2017 (Start): G.f. 3rd column: x^3*(1+x)/((1-x-x^2)*(1+x+x^2)). G.f. 4th column: x^4/((1-x-x^2)*(1+x^2)) =x^4*(1+x)/((1-x-x^2)*(1+x+x^2+x^3)). G.f. 5th column: x^5*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4)). G.f. 6th column: x^6/((1-x-x^2)*(1+x+x^2)*(1-x+x^2)) = x^6*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5)). G.f. 7th column: x^6*(1+x)/(1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6)). G.f. 8th column: x^8/((1-x-x^2)*(1+x^2)*(1+x^4)) = x^8*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6+x^7)). Conjecture (by extrapolating): G.f. kth column: x^k*(1-x^2)/((1-x-x^2)*(1-x^k)). G.f. (1-x^2)/(1-x-x^2)*sum_{i>=1} (x*y)^i/(1-x^i) = (1-x^2)/(1-x-x^2)*A051731(x,y). (End) EXAMPLE Table begins: 1, 2,1, 3,1,1, 5,2,1,1, 8,3,1,1,1, 13,5,3,1,1,1, 21,8,4,2,1,1,1, 34,13,6,4,2,1,1,1, 55,21,11,6,3,2,1,1,1, 89,34,17,9,6,3,2,1,1,1, MAPLE A178534 := proc(n, k)     option remember;     if k= 1 then         combinat[fibonacci](n+1) ;     elif k > n then         0 ;     else         add(procname(n-i, k-1), i=1..k-1)-add(procname(n-i, k), i=1..k-1) ;     end if; end proc: seq(seq(A178534(n, k), k=1..n), n=1..12) ; # R. J. Mathar, Oct 28 2010 PROG (Excel) =if(column()=1; if(row()=1; 1; if(row()=2; 2; indirect(address(row()-1; 1))+indirect(address(row()-2; 1)))); if(row()>=column(); sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-column()+column()-1; column()-1; 4); 4))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-column()+column()-1; column(); 4); 4)); 0)) (Python) from sympy.core.cache import cacheit from sympy import fibonacci @cacheit def A(n, k): return fibonacci(n + 1) if k==1 else 0 if k>n else sum([A(n - i, k - 1) for i in range(1, k)]) - sum([A(n - i, k) for i in range(1, k)]) for n in range(1, 13): print([A(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Sep 15 2017 CROSSREFS Cf. 1st column=A000045(n+1), 2nd=A000045, 3rd=A093040, 4th=A006498. Matrix inverse of A178535. Sequence in context: A026807 A179045 A106740 * A110619 A191861 A129761 Adjacent sequences:  A178531 A178532 A178533 * A178535 A178536 A178537 KEYWORD nonn,tabl,changed AUTHOR Mats Granvik, May 29 2010 STATUS approved

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