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A001303
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Stirling numbers of first kind, s(n+3, n), negated.
(Formerly M4258 N1779)
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17
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6, 50, 225, 735, 1960, 4536, 9450, 18150, 32670, 55770, 91091, 143325, 218400, 323680, 468180, 662796, 920550, 1256850, 1689765, 2240315, 2932776, 3795000, 4858750, 6160050, 7739550, 9642906, 11921175, 14631225, 17836160, 21605760, 26016936, 31154200
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OFFSET
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1,1
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COMMENTS
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a(n) is equal to the sum of the products of each distinct grouping of 3 members of the set {1, 2, 3, ..., n + 2} (a(1) = 1*2*3, a(2) = 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4, a(3) = 1*2*3 + 1*2*4 + 1*2*5 + 1*3*4 + 1*3*5 + 1*4*5 + 2*3*4 + 2*3*5 + 2*4*5 + 3*4*5). - Jeffreylee R. Snow, Sep 23 2013
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
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).
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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a(n) = binomial(n+3, 4)*binomial(n+3, 2).
G.f.: x*(6 + 8*x + x^2)/(1 - x)^7. - Simon Plouffe in his 1992 dissertation
E.g.f. with offset 3: exp(x)*(6*(x^3)/3! + 26*(x^4)/4! + 35*(x^5)/5! + 15*(x^6)/6!). See row k=3 of A112486 for the coefficients [6, 26, 35, 15].
a(n) = (f(n+2, 3)/6!)*Sum_{m=0..min(3, n)} A112486(3,m)*f(6, 3-m)*f(n-1, m), with the falling factorials notation f(n, m):=n*(n-1)*...*(n-(m-1)).
a(n) = A000217(n) * n! / ( 4! * (n-4)! ) [for n > 4 and A000217 = the triangular numbers];
a(n) = ((n+4)! / n! ) ^2 / ( (n+2) * (n+1) * 2*4!);
a(n) = (n-0)^2 * (n-1)^2 * (n-2) * (n-3) / (2*4!). (End)
a(n) = 15*binomial(n+5,6) - 10*binomial(n+4,5) + binomial(n+3,4).
E.g.f. with offset 4: exp(x)*((1/4)*x^4 + (1/6)*x^5 + (1/48)*x^6). (End)
a(n) = (n+3)^2/(n^2-1)*a(n-1), n > 1;
a(n) = 6*Product_{k=2..n} (k+3)^2/(k^2 - 1). (End)
Sum_{n>=1} 1/a(n) = 16*Pi^2/3 - 472/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*Pi^2/3 + 16/9 - 64*log(2)/3. (End)
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MAPLE
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seq(numbperm (n, 2)*numbperm (n, 4)/48, n=4..33); # Zerinvary Lajos, Apr 26 2007
seq(15*binomial(n+2, 6)-10*binomial(n+1, 5)+binomial(n, 4), n=4..30); # Miklos Kristof, Nov 04 2007
-combinat[stirling1](n+3, n) ;
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MATHEMATICA
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Table[-StirlingS1[n + 3, n], {n, 100}] (* T. D. Noe, Jun 27 2012 *)
a[ n_] := n (n + 1) (n + 2)^2 (n + 3)^2 / 48; (* Michael Somos, Sep 04 2017 *)
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PROG
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(Sage) [stirling_number1(n, n-3) for n in range(4, 34)] # Zerinvary Lajos, May 16 2009
(PARI) a(n) = n*(n+1)*(n+2)^2*(n+3)^2/48; \\ Altug Alkan, Aug 29 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
Notation of the polynomial formula edited by R. J. Mathar, Sep 15 2009
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STATUS
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approved
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