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A191242 Reversion of x-x^2-x^3-2*x^4 1
1, 1, 3, 12, 50, 224, 1054, 5121, 25509, 129591, 668811, 3496740, 18481512, 98585788, 530068840, 2869725800, 15630429306, 85589391884, 470905310206, 2601941245750, 14432082902820, 80328808797750, 448527122885700, 2511672193514250 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For the reversion of x - a*x^2 - b*x^3 - c*x^4 (a!=0, b!=0, c!=0) we have

a(n) = sum(k=1,n-1, (sum(j=0..k, a^(-n+3*k-j+1)*b^(n-3*k+2*j-1)*c^(k-j)*binomial(j,n-3*k+2*j-1)*binomial(k,j)))*binomial(n+k-1,n-1))/n, n>1, a(1)=1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

FORMULA

a(n) = sum(k=1..n-1, (sum(j=0..k, binomial(j,n-3*k+2*j-1)*2^(k-j)*binomial(k,j)))*binomial(n+k-1,n-1))/n, n>1, a(1)=1.

MATHEMATICA

a[1] = 1; a[n_] := Sum[Sum[Binomial[j, n - 3k + 2j - 1]*2^(k - j)* Binomial[k, j], {j, 0, k}]*Binomial[n + k - 1, n - 1], {k, 1, n - 1}]/n;

Array[a, 24] (* Jean-Fran├žois Alcover, Jul 23 2018 *)

PROG

(Maxima)

a(n):=sum((sum(binomial(j, n-3*k+2*j-1)*2^(k-j)*binomial(k, j), j, 0, k))*binomial(n+k-1, n-1), k, 1, n-1)/n;

(PARI) x='x+O('x^66); /* that many terms */

Vec(serreverse(x-x^2-x^3-2*x^4)) /* show terms */ /* Joerg Arndt, May 28 2011 */

(MAGMA) [&+[Binomial(i, n-3*k+2*i-1)*2^(k-i)*Binomial(k, i)*Binomial(n+k-1, n-1)/n: k in [0..25], i in [0..n]]: n in [1..25]]; // Vincenzo Librandi, Jul 23 2018

CROSSREFS

Sequence in context: A074547 A151178 A151179 * A105479 A151180 A268650

Adjacent sequences:  A191239 A191240 A191241 * A191243 A191244 A191245

KEYWORD

nonn

AUTHOR

Vladimir Kruchinin, May 28 2011

STATUS

approved

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Last modified January 21 04:53 EST 2020. Contains 331104 sequences. (Running on oeis4.)