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

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A094954 Array T(k,n) read by antidiagonals. G.f.: x(1-x)/(1-kx+x^2), k>1. 27
1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 13, 1, 1, 5, 19, 41, 34, 1, 1, 6, 29, 91, 153, 89, 1, 1, 7, 41, 169, 436, 571, 233, 1, 1, 8, 55, 281, 985, 2089, 2131, 610, 1, 1, 9, 71, 433, 1926, 5741, 10009, 7953, 1597, 1, 1, 10, 89, 631, 3409, 13201, 33461, 47956, 29681 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Also, values of polynomials with coefficients in A098493 (see Fink et al.). See A098495 for negative k.

Number of dimer tilings of the graph S_{k-1} X P_{2n-2}.

REFERENCES

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.

LINKS

Table of n, a(n) for n=1..64.

Elizabeth Wilmer, A note on Stephan's conjecture 87

FORMULA

Recurrence: T(k, 1) = 1, T(k, 2) = k-1, T(k, n) = kT(k, n-1) - T(k, n-2).

For n>3, T(k, n) = [k(k-2) + T(k, n-1)T(k, n-2)] / T(k, n-3).

T(k, n+1) = S(n, k) - S(n-1, k) = U(n, k/2) - U(n-1, k/2), with S, U = Chebyshev polynomials of second kind.

T(k+2, n+1) = Sum[i=0..n, k^(n-i) * C(2n-i, i)] (from comments by Benoit Cloitre).

EXAMPLE

1,1,1,1,1,1,1,1,1,1,1,1,1,1, ...

1,2,5,13,34,89,233,610,1597, ...

1,3,11,41,153,571,2131,7953, ...

1,4,19,91,436,2089,10009,47956, ...

1,5,29,169,985,5741,33461,195025, ...

1,6,41,281,1926,13201,90481,620166, ...

MATHEMATICA

max = 14; row[k_] := Rest[ CoefficientList[ Series[ x*(1-x)/(1-k*x+x^2), {x, 0, max}], x]]; t = Table[ row[k], {k, 2, max+1}]; Flatten[ Table[ t[[k-n+1, n]], {k, 1, max}, {n, 1, k}]] (* Jean-Fran├žois Alcover, Dec 27 2011 *)

PROG

(PARI) T(k, n)=polcoeff(x*(1-x)/(1-k*x+x*x), n)

CROSSREFS

Rows are first differences of rows in array A073134.

Rows 2-14 are A000012, A001519, A079935/A001835, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570. Other rows: A007805 (k=18), A075839 (k=20), A077420 (k=34), A078988 (k=66).

Columns include A028387. Diagonals include A094955, A094956. Antidiagonal sums are A094957.

Sequence in context: A121207 A097084 A143327 * A083064 A204057 A241578

Adjacent sequences:  A094951 A094952 A094953 * A094955 A094956 A094957

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, May 31 2004

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 14 00:21 EST 2017. Contains 295976 sequences.