A081895 -id:A081895 - cp's OEIS frontend

A081896 Third binomial transform of binomial(n+3, 3).

1, 7, 43, 245, 1328, 6944, 35328, 175872, 860160, 4145152, 19726336, 92864512, 433061888, 2002780160, 9193914368, 41926262784, 190052302848, 856845975552, 3843995729920, 17166984282112, 76347338653696, 338237264494592, 1493136790519808, 6569581975961600, 28816000740753408

Offset: 0

Paul Barry, Mar 30 2003

Binomial transform of A081895.

4th binomial transform of (1,3,3,1,0,0,0,0,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256)

Cf. A000292, A081895, A081897.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)^3/(1-4*x)^4)); // G. C. Greubel, Oct 18 2018

LinearRecurrence[{16, -96, 256, -256}, {1, 7, 43, 245}, 50] (* G. C. Greubel, Oct 18 2018 *)
CoefficientList[Series[(1-3x)^3/(1-4x)^4,{x,0,30}],x] (* Harvey P. Dale, Nov 30 2021 *)

x='x+O('x^30); Vec((1-3*x)^3/(1-4*x)^4) \\ G. C. Greubel, Oct 18 2018

a(n) = 4^n*(n^3 + 33*n^2 + 254*n + 384)/384.
G.f.: (1 - 3*x)^3/(1 - 4*x)^4.
E.g.f.: (6 + 18*x + 9*x^2 + x^3)*exp(4*x)/6. - G. C. Greubel, Oct 18 2018

A081897 Fourth binomial transform of binomial(n+3, 3).

Original entry on oeis.org

1, 8, 58, 396, 2595, 16500, 102500, 625000, 3753125, 22250000, 130468750, 757812500, 4365234375, 24960937500, 141796875000, 800781250000, 4498291015625, 25146484375000, 139953613281250, 775756835937500, 4283905029296875, 23574829101562500, 129318237304687500, 707244873046875000

Offset: 0

Paul Barry, Mar 30 2003

Binomial transform of A081895.

5th binomial transform of (1,3,3,1,0,0,0,0,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625)

Cf. A000292, A081895.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)^3/(1-5*x)^4)); // G. C. Greubel, Oct 18 2018

LinearRecurrence[{20, -150, 500, -625}, {1, 8, 58, 396}, 50] (* G. C. Greubel, Oct 18 2018 *)

x='x+O('x^30); Vec((1-4*x)^3/(1-5*x)^4) \\ G. C. Greubel, Oct 18 2018

a(n) = 5^n*(n^3 + 42*n^2 + 407*n + 750)/750.
G.f.: (1 - 4*x)^3/(1 - 5*x)^4.
E.g.f.: (6 + 18*x + 9*x^2 + x^3)*exp(5*x)/6. - G. C. Greubel, Oct 18 2018

cp's OEIS Frontend

A081896 Third binomial transform of binomial(n+3, 3).

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