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A192442
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Coefficient of x^n in the expansion of (1+x^3+x^4)^n.
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2
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1, 0, 0, 3, 4, 0, 15, 42, 28, 84, 360, 495, 715, 2860, 6006, 8463, 23660, 61880, 104244, 220932, 596904, 1201560, 2313003, 5753979, 12983707, 25477100, 57557500, 135227235, 280913490, 600900300, 1395727515, 3046800900, 6447717900, 14540497920, 32572229976, 69844899432
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OFFSET
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0,4
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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Recurrence: 3*n*(3*n-2)*(3*n-1)*(115668*n^3 - 650916*n^2 + 1167723*n - 673723)*a(n) = 2*(n-1)*(231336*n^5 - 1417500*n^4 + 3231306*n^3 - 3349145*n^2 + 1574119*n - 264960)*a(n-1) - 6*(n-2)*(n-1)*(115668*n^4 - 535248*n^3 + 861921*n^2 - 529129*n + 122640)*a(n-2) + 24*(n-2)*(n-1)*(925344*n^4 - 4744656*n^3 + 7608276*n^2 - 4369418*n + 755115)*a(n-3) + 229*(n-3)*(n-2)*(n-1)*(115668*n^3 - 303912*n^2 + 212895*n - 41248)*a(n-4). - Vaclav Kotesovec, Apr 21 2014
a(n) = sum(j=floor(n/4)..floor(n/3), binomial(j,n-3*j)*binomial(n,j)). - Vladimir Kruchinin, Jun 14 2014
G.f.: 1 + x*G'(x)/G(x) where G(x) = 1 + x^3*G(x)^3 + x^4*G(x)^4 = (1/x)*Series_Reversion(x/(1+x^3+x^4)). - Paul D. Hanna, Jun 14 2014
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MATHEMATICA
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Table[Coefficient[(1+x^3+x^4)^n, x, n], {n, 0, 20}] (* Vaclav Kotesovec, Jun 15 2014 *)
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PROG
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(PARI) a(n)=polcoeff((1+x^3+x^4)^n, n);
(PARI) {a(n)=local(G=(1/x)*serreverse(x/(1+x^3+x^4 +x^2*O(x^n)))); polcoeff(1+x*G'/G, n)} \\ Paul D. Hanna, Jun 14 2014
for(n=0, 30, print1(a(n), ", "))
(Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x^3+x^4)^n)[ n+1 ]: n in [0..40] ]; // Vincenzo Librandi, Sep 10 2011
(Maxima) makelist((coeff(expand((1+x^3+x^4)^n), x, n)), n, 0, 40); // Vincenzo Librandi, Sep 10 2011
(Maxima)
a(n):=sum(binomial(j, n-3*j)*binomial(n, j), j, floor(n/4), floor(n/3)); /* Vladimir Kruchinin, Jun 14 2014 */
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CROSSREFS
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Cf. A002426, A192440, A192441.
Sequence in context: A113035 A099447 A078067 * A009126 A322278 A102222
Adjacent sequences: A192439 A192440 A192441 * A192443 A192444 A192445
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KEYWORD
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nonn
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AUTHOR
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Joerg Arndt, Jul 01 2011
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STATUS
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approved
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