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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0. 19
1, 1, 1, 3, 2, 1, 5, 7, 3, 1, 11, 16, 12, 4, 1, 21, 41, 34, 18, 5, 1, 43, 94, 99, 60, 25, 6, 1, 85, 219, 261, 195, 95, 33, 7, 1, 171, 492, 678, 576, 340, 140, 42, 8, 1, 341, 1101, 1692, 1644, 1106, 546, 196, 52, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The g.f. for the row polynomials P(n,x) := sum(a(n,m)*x^m,m=0..n) is 1/(1-(1+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.

The column sequences (without leading zeros) give: A001045(n+1), A073371-9 for m=0..9. Row sums give A002605.

Riordan array (1/(1-x-2x^2),x/(1-x-2x^2)). - Paul Barry, Mar 15 2005

Subtriangle (obtained by dropping the first column) of the triangle given by (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013

The number of ternary words of length n having k letters equal 2 and 0,1 avoid runs of odd lengths. - Milan Janjic, Jan 14 2017

LINKS

Table of n, a(n) for n=0..54.

W. Lang, First 10 rows.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

FORMULA

a(n, m) = sum(binomial(n-k, m)*binomial(n-m-k, k)*2^k, k=0..floor((n-m)/2)) if n>m, else 0.

a(n, m) = (1*(n-m+1)*a(n, m-1)+2*2*(n+m)*a(n-1, m-1))/((1^2+4*2)*m), n>=m>=1, a(n, 0)=A001045(n+1), n>=0, else 0.

a(n, m) = (p(m-1, n-m)*1*(n-m+1)*a(n-m+1)+q(m-1, n-m)*2*(n-m+2)*a(n-m))/(m!*9^m), n>=m>=1, with a(n)=a(n, m=0) := A001045(n+1), else 0; p(k, n) := sum(A(k, l)*n^(k-l), l=0..k) and q(k, n) := sum(B(k, l)*n^(k-l), l=0..k) with the number triangles A(k, m) := A073399(k, m) and B(k, m) := A073400(k, m).

G.f.: 1/(1-(1+2*x)*x)^(m+1), m>=0, for column m (without leading zeros).

T(n, 0)=A001045(n), T(1, 1)=1, T(n, k)=0 if k>n, T(n, k)=T(n-1, k-1)+2T(n-2, k)+T(n-1, k) otherwise. - Paul Barry, Mar 15 2005

G.f.: (1+x)*(2*x-1)/(-1+x+2*x^2+x*y) for the triangle including the 1,0,0,0,0.. column. - R. J. Mathar, Aug 11 2015

EXAMPLE

{1},

{1,1},

{3,2,1},...

(lower triangular matrix n>=m>=0).

The triangle (0, 1, 2, -2, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:

1

0, 1

0, 1, 1

0, 3, 2, 1

0, 5, 7, 3, 1

0, 11, 16, 12, 4, 1

0, 21, 41, 34, 18, 5, 1

- Philippe Deléham, Feb 19 2013

CROSSREFS

Sequence in context: A110712 A065366 A092879 * A208511 A129675 A232206

Adjacent sequences:  A073367 A073368 A073369 * A073371 A073372 A073373

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Aug 02 2002

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 April 26 09:52 EDT 2019. Contains 322472 sequences. (Running on oeis4.)