A081893 -id:A081893 - cp's OEIS frontend

A081892 Second binomial transform of C(n+2,2).

1, 5, 22, 90, 351, 1323, 4860, 17496, 61965, 216513, 747954, 2558790, 8680203, 29229255, 97785144, 325241892, 1076168025, 3544180029, 11622614670, 37967207922, 123587135991, 400980206115, 1297083797172, 4184141281200

Offset: 0

Paul Barry, Mar 30 2003

Binomial transform of A049611(n+1).
2nd binomial transform of C(n+2,2), A000217.
3rd binomial transform of (1,2,1,0,0,0,.....)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A081893.

A right-edge column of triangle A024462.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^2/(1-3*x)^3)); // G. C. Greubel, Oct 18 2018

LinearRecurrence[{9, -27, 27}, {1, 5, 22}, 50] (* G. C. Greubel, Oct 18 2018 *)

x='x+O('x^30); Vec((1-2*x)^2/(1-3*x)^3) \\ G. C. Greubel, Oct 18 2018

a(n) = 3^(n - 2)*(n + 2)*(n + 9)/2 = 3^n*(n^2 + 11*n + 18)/18.
G.f.: (1 - 2*x)^2/(1 - 3*x)^3.
E.g.f.: (2 + 4*x + x^2)*exp(3*x)/2. - G. C. Greubel, Oct 18 2018

A081894 Fourth binomial transform of C(n+2,2).

Original entry on oeis.org

1, 7, 46, 290, 1775, 10625, 62500, 362500, 2078125, 11796875, 66406250, 371093750, 2060546875, 11376953125, 62500000000, 341796875000, 1861572265625, 10101318359375, 54626464843750, 294494628906250, 1583099365234375

Offset: 0

Paul Barry, Mar 30 2003

Binomial transform of A081893.
4th binomial transform of C(n+2,2), A000217.
5th binomial transform of (1,2,1,0,0,0,.....)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-75,125).

Cf. A081907.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)^2/(1-5*x)^3)); // G. C. Greubel, Oct 18 2018

LinearRecurrence[{15, -75, 125}, {1, 7, 46}, 50] (* G. C. Greubel, Oct 18 2018 *)

x='x+O('x^30); Vec((1-4*x)^2/(1-5*x)^3) \\ G. C. Greubel, Oct 18 2018

a(n) = 5^n*(n^2 + 19*n + 50)/50.
G.f.: (1 - 4*x)^2/(1 - 5*x)^3.
E.g.f.: (2 + 4*x + x^2)*exp(5*x)/2. - G. C. Greubel, Oct 18 2018

cp's OEIS Frontend

A081892 Second binomial transform of C(n+2,2).

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