A323618 Expansion of e.g.f. (1 + x)*log(1 + x)*(2 + log(1 + x))/2.

0, 1, 2, -1, 1, -1, -2, 34, -324, 2988, -28944, 300816, -3371040, 40710240, -528439680, 7348717440, -109109064960, 1723814265600, -28888702617600, 512030734387200, -9572240647065600, 188274945999974400, -3887144020408320000, 84062926436751360000, -1900475323780239360000

Offset: 0

Author

Ilya Gutkovskiy, Jan 20 2019

Keywords

• sign

Crossrefs

Cf. A000217, A045406, A048994, A059606, A081052.

Programs

Magma

[(&+[StirlingFirst(n,k)*Binomial(k+1,2): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019

Maple

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):
map(f, [$0..30]); # Robert Israel, Jan 20 2019

Mathematica

nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] (2 + Log[1 + x])/2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k (k + 1)/2, {k, 0, n}], {n, 0, 24}]
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 *)

PARI

{a(n) = sum(k=0,n, stirling(n,k,1)*binomial(k+1,2))};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019

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

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000217(k).

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

a(n) = (5-2*n)*a(n-1) - (n-3)^2*a(n-2) for n >= 4. - Robert Israel, Jan 20 2019