0,4

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.

T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, etc.).

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

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

Equals A049310 * A007318 as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 28 2007

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

Row sums are A000129. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 09 2008

Weisstein, Eric W. "Morgan-Voyce Polynomials." http://mathworld.wolfram.com/Morgan-VoycePolynomials.html

T. Mansour, Generalization of some identities involving the Fibonacci numbers

P. Moree, Convoluted convolved Fibonacci numbers

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.

G.f.: 1/(1 - y - yz - y^2).

G.f. for k-th column: x/(1-x-x^2)^k.

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

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

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

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

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

1; 1,1; 2,2,1; 3,5,3,1; 5,10,9,4,1; etc.

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].

Triangle begins:

..................{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},

........{21,71,111,105,65,27,7,1},

......{34,130,233,256,190,98,35,8,1},

....{55,235,474,594,511,315,140,44,9,1},

{89,420,942,1324,1295,924,490,192,54,10,1}

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

(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)

(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)

A038112(n)=T(2n, n). A038137 is reflected version. Maximal row entries: A038149.

Diagonal differences are in A055830. Vertical sums are in A091186.

Cf. A049310.

Cf. A000129.

Sequence in context: A081572 A144287 A106196 this_sequence A139375 A106198 A054336

Adjacent sequences: A037024 A037025 A037026 this_sequence A037028 A037029 A037030

easy,nonn,tabl

Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 01 1999

Examples from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 27 2004