A006637 Expansion of (2 - x)^4/(1 - x)^8.
16, 96, 344, 952, 2241, 4712, 9108, 16488, 28314, 46552, 73788, 113360, 169507, 247536, 354008, 496944, 686052, 932976, 1251568, 1658184, 2172005, 2815384, 3614220, 4598360, 5802030, 7264296, 9029556, 11148064, 13676487, 16678496, 20225392, 24396768, 29281208
Offset: 0
References
- H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
Programs
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Magma
A006637:= func< n | (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/5040 >; [A006637(n): n in [0..40]]; // G. C. Greubel, Sep 03 2025
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Mathematica
Table[(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/7!, {n,0,40}] (* G. C. Greubel, Sep 03 2025 *) -
SageMath
def A006637(n): return (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)//5040 print([A006637(n) for n in range(41)]) # G. C. Greubel, Sep 03 2025
Formula
G.f.: (2-x)^4/(1-x)^8. - Sean A. Irvine, May 31 2017
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 18 2022
From G. C. Greubel, Sep 03 2025: (Start)
a(n) = Sum_{k=0..4} binomial(4, k)*binomial(n+k+3, k+3).
a(n) = (1/7!)*(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2 + 31*n + 192).
E.g.f.: (1/7!)*(80640 + 403200*x + 423360*x^2 + 161280*x^3 + 27090*x^4 + 2142*x^5 + 77*x^6 + x^7)*exp(x). (End)
Extensions
a(6) and a(8) corrected and more terms from Sean A. Irvine, May 31 2017
New name by G. C. Greubel, Sep 03 2025
Comments