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%I #34 Apr 14 2022 05:34:43
%S 1,63,966,7770,42525,179487,627396,1899612,5135130,12662650,28936908,
%T 62022324,125854638,243577530,452329200,809944464,1404142047,
%U 2364885369,3880739170,6220194750,9759104355,15015551265,22693687380,33738295500,49402080000,71327958156
%N Sixth diagonal of the Stirling2 triangle A048993 and sixth column of triangle A008278.
%H T. D. Noe, <a href="/A112494/b112494.txt">Table of n, a(n) for n = 6..1000</a>
%H S. Butler, P. Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq. 13 (2010), 10.4.4, page 5.
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F a(n) = Stirling2(n, n-5) with Stirling2(n, m)=A048993(n, m). a(n) = A008278(n+5, 6).
%F a(n) = sum(A008517(5, m+1)*binomial(n+5-m, 2*5), m=0..4) from the o.g.f. See p. 257 eq. (6.43) of the R. L. Graham et al. book quoted in A008517.
%F O.g.f.: x*sum(A008517(5, m+1)*x^m, m=0..4)/(1-x)^11 with the fifth row [1, 52, 328, 444, 120] of the second-order Eulerian triangle A008517.
%F E.g.f. with offset n=-4: exp(x)*sum(A112493(5, m)*(x^(m+5))/(m+5)!, m=0..5) with the k=5 row [1, 57, 546, 1750, 2205, 945] of triangle A112493.
%F a(n) = sum(A112493(5, m)*binomial(n+4, 5+m), m=0..5) from the e.g.f. (coefficients from A112493(5, m) are [1, 57, 546, 1750, 2205, 945]).
%F With an offset of 1 the o.g.f. is D^5(x/(1-x)), where D is the operator x/(1-x)*d/dx. - _Peter Bala_, Jul 02 2012
%F G.f.: x^6*(1 + 52*x + 328*x^2 + 444*x^3 + 120*x^4) / (1 - x)^11. - _Colin Barker_, Nov 04 2017
%t Table[StirlingS2[n, n-5], {n, 6, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)
%o (Sage) [stirling_number2(n,n-5) for n in range(6, 30)] # _Zerinvary Lajos_, May 16 2009
%o (PARI) for(n=6,50, print1(stirling(n,n-5,2), ", ")) \\ _G. C. Greubel_, Oct 22 2017
%o (PARI) Vec(x^6*(1 + 52*x + 328*x^2 + 444*x^3 + 120*x^4) / (1 - x)^11 + O(x^40)) \\ _Colin Barker_, Nov 04 2017
%Y Cf. A008517, A112493.
%Y Cf. A001298 (fifth diagonal, resp. column).
%K nonn,easy
%O 6,2
%A _Wolfdieter Lang_, Oct 14 2005
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