A192440 -id:A192440 - cp's OEIS frontend

A192441 Coefficient of x^(2*n) in the expansion of (1 + x^3 + x^4)^n.

1, 0, 2, 3, 6, 30, 35, 210, 350, 1344, 3402, 9240, 29139, 72072, 231660, 603603, 1814670, 5095376, 14507324, 42401502, 118974466, 349305120, 990073812, 2877816304, 8272748675, 23852438880, 69116072950, 198980348385, 577566713520, 1667118322590, 4834810467135

Offset: 0

Joerg Arndt, Jul 01 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A002426, A192440.

P:=PolynomialRing(Integers()); [ Coefficients((1+x^3+x^4)^n)[ 2*n+1 ]: n in [0..30] ]; // Bruno Berselli, Jul 01 2011

makelist((coeff(expand((1+x^3+x^4)^n), x, 2*n)), n, 0, 30); /* Bruno Berselli, Jul 01 2011 */

PARI
```
a(n)=polcoeff((1+x^3+x^4)^n,2*n);
```

a(n) = Sum_{k=ceiling(n/2)..floor(2*n/3)} binomial(n,k)*binomial(k,2*n-3*k). - R. J. Mathar, Jul 01 2011

A192442 Coefficient of x^n in the expansion of (1+x^3+x^4)^n.

Original entry on oeis.org

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

Offset: 0

Joerg Arndt, Jul 01 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A002426, A192440, A192441.

P:=PolynomialRing(Integers()); [ Coefficients((1+x^3+x^4)^n)[ n+1 ]: n in [0..40] ]; // Vincenzo Librandi, Sep 10 2011

Table[Coefficient[(1+x^3+x^4)^n, x, n],{n,0,20}] (* Vaclav Kotesovec, Jun 15 2014 *)

makelist((coeff(expand((1+x^3+x^4)^n), x, n)), n, 0, 40); /* Vincenzo Librandi, Sep 10 2011 */

a(n):=sum(binomial(j,n-3*j)*binomial(n,j),j,floor(n/4),floor(n/3)); /* Vladimir Kruchinin, Jun 14 2014 */

PARI
```
a(n)=polcoeff((1+x^3+x^4)^n,n);
```

{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),", "))

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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A192441 Coefficient of x^(2*n) in the expansion of (1 + x^3 + x^4)^n.

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A192442 Coefficient of x^n in the expansion of (1+x^3+x^4)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

Maxima

PARI

PARI

Formula