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 A229036 G.f.: Sum_{n>=0} (3*n-1)^n * x^n / (1 + (3*n-1)*x)^n. 0
 1, 2, 21, 270, 4212, 77760, 1662120, 40415760, 1102248000, 33331979520, 1107097891200, 40069801094400, 1569793384051200, 66185883219456000, 2988292627358438400, 143855017177487616000, 7355369573944584192000, 398090614491857903616000, 22737098558477268725760000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally, if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n, then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2) so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1. LINKS FORMULA a(n) = (3*n+1) * 3^(n-1) * n!/2 for n>0 with a(0)=1. E.g.f.: (2 - 8*x + 15*x^2)/(2*(1-3*x)^2). EXAMPLE O.g.f.: A(x) = 1 + 2*x + 21*x^2 + 270*x^3 + 4212*x^4 + 77760*x^5 +... where A(x) = 1 + 2*x/(1+2*x) + 5^2*x^2/(1+5*x)^2 + 8^3*x^3/(1+8*x)^3 + 11^4*x^4/(1+11*x)^4 + 14^5*x^5/(1+14*x)^5 +... E.g.f.: E(x) = 1 + 2*x + 21*x^2/2! + 270*x^3/3! + 4212*x^4/4! + 77760*x^5/5! +... where E(x) =  1 + 2*x + 21/2*x^2 + 45*x^3 + 351/2*x^4 + 648*x^5 + 4617/2*x^6 +... which is the expansion of: (2 - 8*x + 15*x^2) / (2 - 12*x + 18*x^2). MAPLE a:=series(add((3*n-1)^n*x^n/(1+(3*n-1)*x)^n, n=0..100), x=0, 19): seq(coeff(a, x, n), n=0..18); # Paolo P. Lava, Mar 27 2019 MATHEMATICA Join[{1}, Table[(3n+1)3^(n-1) n!/2, {n, 20}]] (* Harvey P. Dale, Feb 10 2015 *) PROG (PARI) {a(n)=polcoeff( sum(m=0, n, ((3*m-1)*x)^m / (1 + (3*m-1)*x +x*O(x^n))^m), n)} for(n=0, 20, print1(a(n), ", ")) (PARI) {a(n) = if(n==0, 1, (3*n+1)*3^(n-1)*n!/2)} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A229039, A187735, A187738, A187739, A221160, A221161, A187740. Sequence in context: A105712 A087677 A266969 * A097627 A094797 A234525 Adjacent sequences:  A229033 A229034 A229035 * A229037 A229038 A229039 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 11 2013 STATUS approved

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Last modified August 10 19:52 EDT 2020. Contains 336381 sequences. (Running on oeis4.)