login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A122075 Coefficients of a generalized Pell-Lucas polynomial read by rows. 7
1, 2, 1, 3, 3, 1, 5, 7, 4, 1, 8, 15, 12, 5, 1, 13, 30, 31, 18, 6, 1, 21, 58, 73, 54, 25, 7, 1, 34, 109, 162, 145, 85, 33, 8, 1, 55, 201, 344, 361, 255, 125, 42, 9, 1, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 144, 655, 1416, 1918, 1806, 1239, 630, 236, 63, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
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

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

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

See A055830 for another version.

Sequence in context: A061315 A144265 A209416 * A185675 A153341 A127119

Adjacent sequences:  A122072 A122073 A122074 * A122076 A122077 A122078

KEYWORD

easy,nonn,tabl

AUTHOR

R. J. Mathar, Oct 16 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 20 05:25 EDT 2019. Contains 326139 sequences. (Running on oeis4.)