A348064
Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).
Original entry on oeis.org
1, -2, 25, -75, 1099, -4340, 79064, -382060, 8550916, -48306984, 1303568760, -8346754416, 266955481584, -1894529909376, 70785236377728, -547468189825536, 23610353987137536, -196402650598402560, 9679304091074250240, -85687212859582878720, 4785340778000524477440
Offset: 3
-
a(n) = n!*polcoef(sum(k=3, n, binomial(x, k)), 3);
-
N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^3/(6*(1-x))))
-
from sympy.abc import x
from sympy import ff, expand
def A348064(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**3) for k in range(3,n+1)) # Chai Wah Wu, Sep 27 2021
A081052
Difference of Stirling numbers of the first kind.
Original entry on oeis.org
0, 1, -4, 17, -85, 499, -3388, 26200, -227708, 2199276, -23382216, 271461816, -3418002432, 46399476096, -675622445184, 10504980616320, -173726527230720, 3045008035203840, -56389237652344320, 1100174877158791680, -22556707790402304000, 484876713643386624000
Offset: 0
-
Table[StirlingS1[n,2]-StirlingS1[n,3],{n,30}] (* Harvey P. Dale, May 02 2012 *)
-
for(n=1, 22, print1(stirling(n, 2) - stirling(n, 3),", ")) \\ Indranil Ghosh, Mar 11 2017
A346945
Expansion of e.g.f. log( 1 + log(1 + x)^3 / 3! ).
Original entry on oeis.org
1, -6, 35, -235, 1834, -16352, 164044, -1830630, 22513326, -302700926, 4419167532, -69637654996, 1178377833424, -21315571470320, 410529985172400, -8388475139138320, 181270810764205440, -4130796696683135280, 99008773205008777760, -2490134250475836315120
Offset: 3
-
nmax = 22; CoefficientList[Series[Log[1 + Log[1 + x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS1[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 22}]
A231121
Denominators of coefficients of expansion of arctan(x)^3.
Original entry on oeis.org
1, 1, 15, 945, 175, 17325, 23648625, 1576575, 7309575, 1283268987, 3360942585, 1932541986375, 135664447443525, 218461268025, 242856109621125, 27604644460267875, 4479480941961650625, 1151866527932995875, 31580724596338947904875, 809762169136896100125, 4742892704944677157875
Offset: 0
-
a[n_] := SeriesCoefficient[ArcTan[x]^3, {x, 0, 2*n+3}] // Denominator
(* or *) a[n_] := 3*Sum[2^(i-2)*Binomial[2*(n+1), i-1]*StirlingS1[i, 3]/i!, {i, 3, 2n+3}] // Denominator; Table[a[n], {n, 0, 20}] (* from the formula given by Ruperto Corso in A002429 *)
Take[Denominator[CoefficientList[Series[ArcTan[x]^3,{x,0,50}],x] ], {4,-1,2}] (* Harvey P. Dale, Apr 07 2017 *)
Showing 1-4 of 4 results.