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A179848
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Expansion of series reversion of generating function for triangular numbers.
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10
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0, 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, -50067108, 300830572, -1822766520, 11124755664, -68328754959, 422030545335, -2619631042665, 16332922290300, -102240109897695, 642312451217745, -4048514844039120, 25594403741131680
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A(x) satisfies A(x) = x * (1 - A(x))^3.
G.f.: 1 - sinh( arcsinh( sqrt( 27*x/4 ) ) / 3 ) / sqrt( 3*x/4 ).
D-finite with recurrence +2*n*(2*n+1)*a(n) +3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 24 2023
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EXAMPLE
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G.f. = x - 3*x^2 + 12*x^3 - 55*x^4 + 273*x^5 - 1428*x^6 + 7752*x^7 - 43263*x^8 + ...
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MAPLE
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a:= n-> coeff(series(RootOf(A=x*(1-A)^3, A), x, n+1), x, n):
# Using function CompInv from A357588.
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MATHEMATICA
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CoefficientList[Series[1 - Sinh[ArcSinh[Sqrt[27*x/4]]/3]/Sqrt[3*x/4], {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, -(-1)^n * (3*n)! / (n! * (2*n+1)!) )};
(PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x / (1 - x)^3 + x * O(x^n) ), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for( k = 0, n, A = x * (1 - A)^3 ); polcoeff( A, n ))};
(Magma) [n le 0 select 0 else (-1)^(n+1)*Factorial(3*n)/( Factorial(n)* Factorial(2*n+1)): n in [0..30]]; // G. C. Greubel, Aug 14 2018
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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