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Search: id:A094954
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| A094954 |
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Array T(k,n) read by antidiagonals. G.f.: x(1-x)/(1-kx+x^2), k>1. |
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24
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| 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; listen)
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OFFSET
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1,5
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COMMENT
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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}.
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REFERENCES
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A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
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LINKS
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Elizabeth Wilmer, A note on Stephan's conjecture 87
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FORMULA
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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).
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EXAMPLE
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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, ...
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PROGRAM
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(PARI) T(k, n)=polcoeff(x*(1-x)/(1-k*x+x*x), n)
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CROSSREFS
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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 this_sequence A083064 A112338 A111672
Adjacent sequences: A094951 A094952 A094953 this_sequence A094955 A094956 A094957
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan (ralf(AT)ark.in-berlin.de), May 31 2004
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