Search: id:A037027
Results 1-1 of 1 results found.

%I A037027
%S A037027 1,1,1,2,2,1,3,5,3,1,5,10,9,4,1,8,20,22,14,5,1,13,38,51,40,20,6,1,21,71,
%T A037027 111,105,65,27,7,1,34,130,233,256,190,98,35,8,1,55,235,474,594,511,315,
%U A037027 140,44,9,1,89,420,942,1324,1295,924,490,192,54,10,1,144,744,1836
%N A037027 Skew Fibonacci-Pascal triangle read by rows.
%C A037027 Row sums form Pell numbers A000129, T(n,0) forms Fibonacci numbers A000045, 
               T(n,1) forms A001629. T(n+k,n-k) is polynomial sequence of degree 
               k.
%C A037027 T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, etc.).
%C A037027 As a Riordan array, this is (1/(1-x-x^2),x/(1-x-x^2)). An interesting 
               factorization is (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x)) [abs(A049310) 
               times A007318]. Diagonal sums are the Jacobsthal numbers A001045(n+1). 
               - Paul Barry (pbarry(AT)wit.ie), Jul 28 2005
%C A037027 T(n,k) = T'(n+1,k+1), T' given by [0, 1, 1, -1, 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 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 19 2005
%C A037027 Equals A049310 * A007318 as infinite lower triangular matrices. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Oct 28 2007
%C A037027 This triangle may also be obtained from the coefficients of the Morgan-Voyce 
               polynomials defined by: Mv(x, n) = (x + 1)*Mv(x, n - 1) + Mv(x, n 
               - 2). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 09 2008
%C A037027 Row sums are A000129. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Apr 09 2008
%D A037027 Weisstein, Eric W. "Morgan-Voyce Polynomials." http://mathworld.wolfram.com/
               Morgan-VoycePolynomials.html
%H A037027 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0301157">Generalization 
               of some identities involving the Fibonacci numbers</a>
%H A037027 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved 
               Fibonacci numbers</a>
%F A037027 T(n, m) = T'(n-1, m)+T'(n-2, m)+T'(n-1, m-1), where T'(n, m) = T(n, m) 
               for n >= 0 and 0< = m< = n and T'(n, m) = 0 otherwise.
%F A037027 G.f.: 1/(1 - y - yz - y^2).
%F A037027 G.f. for k-th column: x/(1-x-x^2)^k.
%F A037027 T(n, m)= sum(binomial(m+k, m)*binomial(k, n-k-m), k=0..n-m), n>=m>=0, 
               else 0. Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Jun 17 2002
%F A037027 T(n, m) = ((n-m+1)*T(n, m-1) + 2*(n+m)*T(n-1, m-1))/(5*m), n >= m >= 
               1; T(n, 0)= A000045(n+1); T(n, m)= 0 if n<m. Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)\
               , Apr 12 2000
%F A037027 Chebyshev coefficient triangle (abs(A049310)) times Pascal's triangle 
               (A007318) as product of lower triangular matrices. T(n, k)=sum{k=0..n, 
               C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}. - Paul Barry (pbarry(AT)wit.ie), 
               Dec 22 2004
%F A037027 Let R(n) = n-th row polynomial in x, with R(0)=1, then R(n+1)/R(n) equals 
               the continued fraction [1+x;1+x, ...(1+x) occurring (n+1) times..., 
               1+x] for n>=0. - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 27 2004
%F A037027 T(n,k)=sum{j=0..n, C(n-j,j)C(n-2j,k)}; in Egorychev notation, T(n,k)=res_w(1-w-w^2)^(-k-1)*w^(-n+k+1). 
               - Paul Barry (pbarry(AT)wit.ie), Sep 13 2006
%F A037027 Sum_{k, 0<=k<=n}T(n,k)*x^k = A000045(n+1), A000129(n+1), A006190(n+1), 
               A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), 
               A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), 
               A041113(n) for n = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. 
               [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 29 2009]
%e A037027 1; 1,1; 2,2,1; 3,5,3,1; 5,10,9,4,1; etc.
%e A037027 Ratio of row polynomials R(3)/R(2) = (3+5*x+3*x^2+x^3)/(2+2*x+x^2) = 
               [1+x;1+x,1+x].
%e A037027 Triangle begins:
%e A037027 ..................{1},
%e A037027 .................{1,1},
%e A037027 ................{2,2,1},
%e A037027 ...............{3,5,3,1},
%e A037027 ..............{5,10,9,4,1},
%e A037027 ............{8,20,22,14,5,1},
%e A037027 ..........{13,38,51,40,20,6,1},
%e A037027 ........{21,71,111,105,65,27,7,1},
%e A037027 ......{34,130,233,256,190,98,35,8,1},
%e A037027 ....{55,235,474,594,511,315,140,44,9,1},
%e A037027 {89,420,942,1324,1295,924,490,192,54,10,1}
%t A037027 Clear[Mv, a] Mv[x, -1] = 0; Mv[x, 0] = 1; Mv[x, 1] = 1 + x; Mv[x_, n_] 
               := Mv[x, n] = (x + 1)*Mv[x, n - 1] + Mv[x, n - 2]; Table[ExpandAll[Mv[x, 
               n]], {n, 0, 10}]; a = Table[CoefficientList[Mv[x, n], x], {n, 0, 
               10}]; Flatten[a] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 
               09 2008
%o A037027 (PARI) T(n,k)=if(k<0|k>n,0,if(n==0&k==0,1,T(n-1,k)+T(n-1,k-1)+T(n-2,k))) 
               (from Michael Somos)
%o A037027 (PARI) T(n,k)=if(n<k|k<0,0,polcoeff(contfracpnqn(vector(n,i,1+x))[1,1],
               k,x)) (from Paul D. Hanna)
%Y A037027 A038112(n)=T(2n, n). A038137 is reflected version. Maximal row entries: 
               A038149.
%Y A037027 Diagonal differences are in A055830. Vertical sums are in A091186.
%Y A037027 Cf. A049310.
%Y A037027 Cf. A000129.
%Y A037027 Sequence in context: A081572 A144287 A106196 this_sequence A139375 A106198 
               A054336
%Y A037027 Adjacent sequences: A037024 A037025 A037026 this_sequence A037028 A037029 
               A037030
%K A037027 easy,nonn,tabl,new
%O A037027 0,4
%A A037027 Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 01 1999
%E A037027 Examples from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 27 2004

Search completed in 0.002 seconds