A081894 -id:A081894 - cp's OEIS frontend

A081893 Third binomial transform of C(n+2,2).

1, 6, 33, 172, 864, 4224, 20224, 95232, 442368, 2031616, 9240576, 41680896, 186646528, 830472192, 3674210304, 16173236224, 70866960384, 309237645312, 1344324763648, 5823975653376, 25151328485376, 108301895335936

Offset: 0

Paul Barry, Mar 30 2003

Binomial transform of A081892.
3rd binomial transform of C(n+2,2), A000217.
4th 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 (12,-48,64).

Cf. A081894.

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

LinearRecurrence[{12, -48, 64}, {1, 6, 33}, 50] (* G. C. Greubel, Oct 18 2018 *)

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

a(n) = 4^n*(n^2 + 15*n + 32)/32.
G.f.: (1 - 3*x)^2/(1 - 4*x)^3.
E.g.f.: (2 + 4*x + x^2)*exp(4*x)/2. - G. C. Greubel, Oct 18 2018

A081907 Fifth binomial transform of binomial(n+2, 2).

Original entry on oeis.org

1, 8, 61, 450, 3240, 22896, 159408, 1096416, 7464960, 50388480, 337602816, 2247326208, 14874679296, 97955205120, 642150789120, 4192482779136, 27270729105408, 176789554200576, 1142549512519680, 7363096858460160, 47326939807481856, 303461150525227008, 1941420131673440256

Offset: 0

Paul Barry, Mar 30 2003

Binomial transform of A081894.

6th 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 (18,-108,216).

Cf. A000217, A081894.

List([1..20],n->6^(n-1)*(n^2+21*n+50))/72; # Muniru A Asiru, Oct 18 2018

[6^n*(n^2 +23*n +72)/72: n in [0..50]]; // G. C. Greubel, Oct 17 2018

seq(coeff(series((1-5*x)^2/(1-6*x)^3,x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 18 2018

Table[6^n*(n^2+23*n+72)/72, {n,0,50}] (* or *) LinearRecurrence[{18,-108, 216}, {1, 8, 61}, 50] (* G. C. Greubel, Oct 17 2018 *)

vector(50, n, n--; 6^n*(n^2 +23*n +72)/72) \\ G. C. Greubel, Oct 17 2018

a(n) = 6^n*(n^2 + 23*n + 72)/72.
G.f.: (1-5*x)^2/(1-6*x)^3.
E.g.f.: (2 + 4*x + x^2)*exp(6*x)/2. - G. C. Greubel, Oct 17 2018

cp's OEIS Frontend

A081893 Third binomial transform of C(n+2,2).

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