OFFSET
0,2
COMMENTS
A122075 is jointly generated with A037027 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012
Subtriangle of the triangle T(n,k) given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 11 2012
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Tian-Xiao He and Peter J.-S. Shiue, Identities for linear recursive sequences of order 2, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
Y. Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370.
FORMULA
T(n,k)=sum_(j=0..n-k+1) binomial(n-k-j+1,j)*binomial(n-j,k).
sum_(k>=0) T(n-k,k)=2^n.
sum_(k>=0) (-1)^k T(n-k,k)=2-delta(0,n).
G.f.: -(1+x)/(-1+x*y+x+x^2). - R. J. Mathar, Aug 11 2015
EXAMPLE
Triangle begins
1
2 1
3 3 1
5 7 4 1
8 15 12 5 1
13 30 31 18 6 1
A055830 = (1, 1, -1, 0, 0, 0, ...) DELTA ((0, 1, 0, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 3, 1, 0
5, 7, 4, 1, 0
8, 15, 12, 5, 1, 0
13, 30, 31, 18, 6, 1, 0
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A122075 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A037027 *)
(* Clark Kimberling, Mar 05 2012 *)
CoefficientList[CoefficientList[Series[-(1 + x)/(-1 + x*y + x + x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Dec 24 2017 *)
PROG
(PARI) T(n, k)={ sum(j=0, n-k+1, binomial(n-k-j+1, j)*binomial(n-j, k)) ; } { nmax=10 ; for(n=0, nmax, for(k=0, n, print1(T(n, k), ", ") ; ); ); }
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Oct 16 2006
STATUS
approved