A081905 Third binomial transform of binomial(n+6, 6).

1, 10, 79, 552, 3567, 21810, 127905, 725820, 4009920, 21664000, 114840064, 598865920, 3078537216, 15626600448, 78431059968, 389685706752, 1918516592640, 9367021682688, 45387134009344, 218388081147904, 1044061452500992, 4961718019031040, 23449374679891968, 110252343064264704

Offset: 0

Author

Paul Barry, Mar 31 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A081904.

4th binomial transform of (1,6,15,20,15,6,1,0,0,0,...).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (28,-336,2240,-8960,21504,-28672,16384).

Crossrefs

Cf. A000579, A081904, A081906.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)^6/(1-4*x)^7)); // G. C. Greubel, Oct 17 2018

Mathematica

CoefficientList[Series[(1-3x)^6/(1-4x)^7,{x,0,20}],x] (* or *) LinearRecurrence[{28,-336,2240,-8960,21504,-28672,16384},{1,10,79,552,3567,21810,127905},20] (* Harvey P. Dale, Aug 14 2014 *)

PARI

x='x+O(x^30); Vec((1-3*x)^6/(1-4*x)^7) \\ G. C. Greubel, Oct 17 2018

Formula

a(n) = 4^n*(n^6 + 129*n^5 + 5845*n^4 + 115215*n^3 + 993874*n^2 + 3308616*n + 2949120)/2949120.

G.f.: (1-3*x)^6/(1-4*x)^7.

a(n) = 28*a(n-1) - 336*a(n-2) + 2240*a(n-3) - 8960*a(n-4) + 21504*a(n-5) - 28672*a(n-6) + 16384*a(n-7); a(0)=1, a(1)=10, a(2)=79, a(3)=552, a(4)=3567, a(5)=21810, a(6)=127905. - Harvey P. Dale, Aug 14 2014

E.g.f.: (720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(4*x) / 720. - G. C. Greubel, Oct 17 2018