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A001875
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Convolved Fibonacci numbers.
(Formerly M4412 N1865)
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5
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1, 7, 35, 140, 490, 1554, 4578, 12720, 33705, 85855, 211519, 506408, 1182650, 2702350, 6056850, 13343820, 28947240, 61926900, 130814600, 273163100, 564415390, 1154933230, 2342193350, 4710707400, 9401674275, 18629923053, 36670044621, 71728832280, 139485074370
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OFFSET
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0,2
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COMMENTS
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a(n) = (((-i)^n)/6!)*((d^5/dx^5) S(n+6,x))|_{x=i}. Sixth derivative of Chebyshev S(n+6,x) polynomials evaluated at x=i (imaginary unit) multiplied by ((-i)^n)/6!. See A049310 for the S-polynomials. - Wolfdieter Lang, Apr 04 2007
a(n) = number of weak compositions of n in which exactly 6 part are 0 and all other parts are either 1 or 2. - Milan Janjic, Jun 28 2010
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REFERENCES
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J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
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LINKS
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M. S. Waterman, Home Page (contains copies of his papers)
Index entries for linear recurrences with constant coefficients, signature (7, -14, -7, 49, -14, -77, 29, 77, -14, -49, -7, 14, 7, 1).
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FORMULA
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G.f.: (1 - x - x^2)^(-7).
a(n) = F''''''(n+6, 1)/6!, i.e., 1/6! times the 6th derivative of the (n+6)th Fibonacci polynomial evaluated at 1. - T. D. Noe, Jan 18 2006
a(n) = Sum_{k=ceiling(n/2)..n} (k+1)*(k+2)*(k+3)*(k+4)*(k+5)*(k+6)*binomial(k,n-k)/720. - Vladimir Kruchinin, Apr 26 2011
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MAPLE
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a:= n-> (Matrix(14, (i, j)-> if (i=j-1) then 1 elif j=1 then [7, -14, -7, 49, -14, -77, 29, 77, -14, -49, -7, 14, 7, 1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=0..22); # Alois P. Heinz, Aug 15 2008
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MATHEMATICA
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nn = 30; CoefficientList[Series[1/(1 - x - x^2)^7, {x, 0, nn}], x] (* T. D. Noe, Aug 10 2012 *)
LinearRecurrence[{7, -14, -7, 49, -14, -77, 29, 77, -14, -49, -7, 14, 7, 1}, {1, 7, 35, 140, 490, 1554, 4578, 12720, 33705, 85855, 211519, 506408, 1182650, 2702350}, 30] (* Harvey P. Dale, Aug 05 2023 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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