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 A296660 Expansion of the e.g.f. exp(-2*x)/(1-4*x). 0
 1, 2, 20, 232, 3728, 74528, 1788736, 50084480, 1602703616, 57697329664, 2307893187584, 101547300251648, 4874270412083200, 253462061428318208, 14193875439985836032, 851632526399150129152, 54504481689545608331264, 3706304754889101366394880 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Binomial self-convolution of sequence A296618. LINKS Table of n, a(n) for n=0..17. FORMULA E.g.f.: exp(-2*x)/(1-4*x). a(n) = Sum_{k=0..n} binomial(n,k)*4^k*k!*(-2)^(n-k). Sum_{k=0..n} binomial(n,k)*2^(n-k)*a(k) = 4^n n!. a(n+1)-4*(n+1)*a(n) = (-2)^(n+1). D-finite with recurrence a(n+2)-(4*n+6)*a(n+1)-8*(n+1)*a(n) = 0. From Vaclav Kotesovec, Dec 18 2017: (Start) a(n) = exp(-1/2) * 4^n * Gamma(n + 1, -1/2). a(n) ~ n! * exp(-1/2) * 4^n. (End) MATHEMATICA CoefficientList[Series[Exp[-2x]/(1-4x), {x, 0, 12}], x]Range[0, 12]! Table[Sum[Binomial[n, k] 4^k k! (-2)^(n-k), {k, 0, n}], {n, 0, 12}] PROG (Maxima) makelist(sum(binomial(n, k)*4^k*k!*(-2)^(n-k), k, 0, n), n, 0, 12); (PARI) x='x+O('x^99); Vec(serlaplace(exp(-2*x)/(1-4*x))) \\ Altug Alkan, Dec 18 2017 CROSSREFS Cf. A001907, A056545, A097820, A296618. Sequence in context: A127110 A361964 A337856 * A197898 A293471 A109106 Adjacent sequences: A296657 A296658 A296659 * A296661 A296662 A296663 KEYWORD nonn AUTHOR Emanuele Munarini, Dec 18 2017 STATUS approved

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Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)