A001875 Convolved Fibonacci numbers.

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

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Comments

a(n) = (((-i)^n)/6!)*((d^5/dx^5) S(n+6,x))|A049310%20for%20the%20S-polynomials.%20-%20_Wolfdieter%20Lang">{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

References

• 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.

Links

• T. D. Noe, Table of n, a(n) for n = 0..500
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers [Corrected annotated scanned copy]
• 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).

Programs

Maple

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

Mathematica

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 *)

Formula

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