A076009 -id:A076009 - cp's OEIS frontend

A076008 Second column of triangle A075504.

1, 27, 567, 10935, 203391, 3720087, 67493007, 1219657095, 21996874431, 396331160247, 7137447668847, 128505439098855, 2313380333315871, 41643387865514007, 749603858371707087, 13493075341822822215

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..1} (A075513(3,m)*exp(9*(m+1)*x)).

Michael De Vlieger, Table of n, a(n) for n = 0..796
Index entries for linear recurrences with constant coefficients, signature (27,-162).

Cf. A001019, A076009.

CoefficientList[Series[1/((1-9x)(1-18x)),{x,0,30}],x] (* or *) LinearRecurrence[{27,-162},{1,27},30] (* Harvey P. Dale, Dec 01 2015 *)

a(n) = A075504(n+2, 2) = (9^n)*S2(n+2, 2) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = -9^n + 2*18^n.
G.f.: 1/((1-9*x)*(1-18*x)).
E.g.f.: (d^2/dx^2)(((exp(9*x)-1)/9)^2)/2! = -exp(9*x) + 2*exp(18*x).
a(0)=1, a(1)=27, a(n) = 27*a(n-1) - 162*a(n-2). - Harvey P. Dale, Dec 01 2015

A076010 Fourth column of triangle A075504.

Original entry on oeis.org

1, 90, 5265, 255150, 11160261, 458810730, 18124795305, 697117731750, 26323112938221, 981154011007170, 36233774365169745, 1329174591745823550, 48521083977375207381, 1764912230785563088410, 64027726517340144702585

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..3} (A075513(4,m)*exp(9*(m+1)*x))/3!.

Michael De Vlieger, Table of n, a(n) for n = 0..641

Cf. A076009, A076011.

With[{m = 4}, Array[9^(# - m) StirlingS2[#, m] &, 15, m]] (* Michael De Vlieger, Dec 24 2017, after Indranil Ghosh at A075504 *)

a(n) = A075504(n+4, 4) = (9^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (-9^n + 24*18^n - 81*27^n + 64*36^n)/3!.
G.f.: 1/Product_{k=1..4} (1 - 9*k*x).
E.g.f.: (d^4/dx^4)(((exp(9*x)-1)/9)^4)/4! = (-exp(9*x) + 24*exp(18*x) - 81*exp(27*x) + 64*exp(36*x))/3!.

cp's OEIS Frontend

A076008 Second column of triangle A075504.

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