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%I #14 Sep 08 2022 08:46:23
%S 0,1,2,-1,1,-1,-2,34,-324,2988,-28944,300816,-3371040,40710240,
%T -528439680,7348717440,-109109064960,1723814265600,-28888702617600,
%U 512030734387200,-9572240647065600,188274945999974400,-3887144020408320000,84062926436751360000,-1900475323780239360000
%N Expansion of e.g.f. (1 + x)*log(1 + x)*(2 + log(1 + x))/2.
%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A000217(k).
%F a(n) ~ -(-1)^n * log(n) * n! / n^2 * (1 + (gamma - 2)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 20 2019
%F a(n) = (5-2*n)*a(n-1) - (n-3)^2*a(n-2) for n >= 4. - _Robert Israel_, Jan 20 2019
%p f:= gfun:-rectoproc({a(n) = (5-2*n)*a(n-1) - (n-3)^2*a(n-2), a(0)=0, a(1)=1, a(2)=2, a(3)=-1}, a(n), remember):
%p map(f, [$0..30]); # _Robert Israel_, Jan 20 2019
%t nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] (2 + Log[1 + x])/2, {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[StirlingS1[n, k] k (k + 1)/2, {k, 0, n}], {n, 0, 24}]
%t Join[{0,1,2,-1}, RecurrenceTable[{a[n]==(5-2*n)*a[n-1]-(n-3)^2*a[n-2], a[2]==2, a[3]==-1}, a, {n,4,25}]] (* _G. C. Greubel_, Feb 07 2019 *)
%o (PARI) {a(n) = sum(k=0,n, stirling(n,k,1)*binomial(k+1,2))};
%o vector(30, n, n--; a(n)) \\ _G. C. Greubel_, Feb 07 2019
%o (Magma) [(&+[StirlingFirst(n,k)*Binomial(k+1,2): k in [0..n]]): n in [0..25]]; // _G. C. Greubel_, Feb 07 2019
%o (Sage) [sum((-1)^(k+n)*stirling_number1(n,k)*binomial(k+1,2) for k in (0..n)) for n in (0..25)] # _G. C. Greubel_, Feb 07 2019
%Y Cf. A000217, A045406, A048994, A059606, A081052.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jan 20 2019
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