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 A195136 a(n) = ((n+1)^(n-1) + (n-1)^(n-1))/2 for n>=1. 3
 1, 2, 10, 76, 776, 9966, 154400, 2803256, 58388608, 1372684090, 35958682112, 1038736032324, 32805006411776, 1124535087475814, 41584800431742976, 1650158470945337584, 69943137585151901696, 3153813559835569475058, 150745204037648268787712, 7613458147995669857352380, 405143549343202022103973888, 22657085569540734204315357022, 1328470689420203636727039918080, 81494507575933974604289943213096, 5220210773193749540624447754469376, 348542314841685116176787263033063466, 24216786265392720787141148530274467840, 1748280517106781152846793195054531026356, 130956723831431687431286364126682302906368, 10164786953127554557192799138093559445158870 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Table of n, a(n) for n=1..30. FORMULA E.g.f.: sinh(x*W(x)) = (W(x) - 1/W(x))/2 where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!. a(n) = Sum_{k=0..floor((n-1)/2)} C(n-1,2*k) * n^(n-2*k-1). EXAMPLE E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 76*x^4/4! + 776*x^5/5! + 9966*x^6/6! + 154400*x^7/7! + 2803256*x^8/8! + 58388608*x^9/9! + 1372684090*x^10/10! +... such that A(x) = sinh(x*W(x)) where W(x) = LambertW(-x)/(-x) begins W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +... and satisfies W(x) = exp(x*W(x)). Also, A(x) = (W(x) - 1/W(x))/2 where 1/W(x) = 1 - x - x^2/2! - 4*x^3/3! - 27*x^4/4! - 256*x^5/5! - 3125*x^6/6! - 46656*x^7/7! - 823543*x^8/8! +...+ -(n-1)^(n-1)*x^n/n! +... MATHEMATICA Join[{1}, Table[((n+1)^(n-1)+(n-1)^(n-1))/2, {n, 2, 30}]] (* Harvey P. Dale, Feb 06 2023 *) PROG (PARI) {a(n)=((n+1)^(n-1) + (n-1)^(n-1))/2} for(n=1, 30, print1(a(n), ", ")) (PARI) {a(n)=sum(k=0, (n-1)\2, binomial(n-1, 2*k)*n^(n-2*k-1))} for(n=1, 30, print1(a(n), ", ")) (PARI) {a(n)=local(W=sum(m=0, n, (m+1)^(m-1)*x^m/m!)+x*O(x^n)); n!*polcoeff(sinh(x*W), n)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A000272, A274278, A274279. Sequence in context: A355110 A088500 A295929 * A294573 A301741 A375876 Adjacent sequences: A195133 A195134 A195135 * A195137 A195138 A195139 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 09 2011 EXTENSIONS Entry revised by Paul D. Hanna, Jun 19 2016 Corrected and extended by Harvey P. Dale, Feb 06 2023 STATUS approved

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Last modified September 12 16:05 EDT 2024. Contains 375853 sequences. (Running on oeis4.)