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 A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k. 10
 1, 3, 21, 171, 1521, 14283, 138909, 1385163, 14072193, 145039923, 1512191781, 15914734443, 168802010001, 1802247516891, 19350710547021, 208783189719531, 2262263134211073, 24604815145831011, 268499713118585781, 2938736789722114731, 32250788066104022961 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ignoring initial term, equals the logarithmic derivative of A206177. Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3. Diagonal of rational functions 1/(1 - x*y + y*z + 2*x*z - 3*x*y*z), 1/(1 + y + z + x*y + y*z + 2*x*z + 3*x*y*z), 1/(1 - x + 2*z + x*y - y*z - 2*x*z + 3*x*y*z), 1/(1 - x - y - z + x*y + y*z + x*z - 3*x*y*z), 1/(1 - x + y + 2*z - x*y + 2*y*z - 2*x*z - 3*x*y*z). - Gheorghe Coserea, Jul 03 2018 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012. FORMULA a(2*3^n) == 3 (mod 9) for n>=0; a(n) == 0 (mod 9) if n/2 > 1 is not a power of 3. Recurrence: (n+3)^2*(3*n+4)*a(n+3) - 3*(9*n^3+57*n^2+116*n+74)*a(n+2) - 3*(27*n^3+144*n^2+252*n+145)*a(n+1) - 27*(3*n+7)*(n+1)^2*a(n) = 0. - Vaclav Kotesovec, Sep 11 2012 a(n) ~ (1+2^(1/3))^2/(2*2^(1/3)*sqrt(3)*Pi) * (3*2^(2/3)+3*2^(1/3)+3)^n/n. - Vaclav Kotesovec, Sep 19 2012 G.f.: hypergeom([1/3, 2/3],[1],54*x^2/(1-3*x)^3)/(1-3*x). - Mark van Hoeij, May 02 2013 a(n) = hypergeom([-n,-n,-n],[1,1], -2). - Peter Luschny, Sep 23 2014 G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 27*x^2 + 9*x - 1)*y'' + (243*x^4 + 378*x^3 + 189*x^2 + 36*x - 2)*y' + 3*(x + 1)*(27*x^2 + 12*x + 2)*y. - Gheorghe Coserea, Jul 01 2018 EXAMPLE L.g.f.: L(x) = 3*x + 21*x^2/2 + 171*x^3/3 + 1521*x^4/4 + 14283*x^5/5 +... Exponentiation equals the g.f. of A206177: exp(L(x)) = 1 + 3*x + 15*x^2 + 93*x^3 + 657*x^4 + 5067*x^5 + 41579*x^6 +... MATHEMATICA Flatten[{1, RecurrenceTable[{(n+3)^2*(3*n+4)*a[n+3]-3*(9*n^3+57*n^2+116*n+74)*a[n+2]-3*(27*n^3+144*n^2+252*n+145)*a[n+1]-27*(3*n+7)*(n+1)^2*a[n]==0, a[1]==3, a[2]==21, a[3]==171}, a, {n, 1, 20}]}] (* Vaclav Kotesovec, Sep 11 2012 *) Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -2], {n, 0, 20}] (* Jean-François Alcover, Oct 25 2019 *) PROG (PARI) {a(n)=sum(k=0, n, binomial(n, k)^3*2^k)} (Sage) A206178 = lambda n: hypergeometric([-n, -n, -n], [1, 1], -2) [Integer(A206178(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 23 2014 CROSSREFS Cf. A206177, A000172, A206180, A216483, A216636. Related to diagonal of rational functions: A268545-A268555. Sequence in context: A132805 A189475 A331328 * A233861 A206397 A247480 Adjacent sequences:  A206175 A206176 A206177 * A206179 A206180 A206181 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 04 2012 EXTENSIONS Minor edits by Vaclav Kotesovec, Mar 31 2014 STATUS approved

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Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)