A081105 -id:A081105

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A079028

a(0) = 1, a(n) = (n + 4)*4^(n-1) for n >= 1.

+10
6

1, 5, 24, 112, 512, 2304, 10240, 45056, 196608, 851968, 3670016, 15728640, 67108864, 285212672, 1207959552, 5100273664, 21474836480, 90194313216, 377957122048, 1580547964928, 6597069766656, 27487790694400, 114349209288704, 474989023199232, 1970324836974592, 8162774324609024

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) = det(M(n)) where M(n) is the n X n matrix defined by m(i,i) = 5, m(i,j) = i/j.

Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 3*m(i-1,j-1).

4th binomial transform of (1,1,0,0,0,0,...). - Paul Barry, Mar 07 2003

Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the complete graph K_n (see A235113). Example: a(1)=5; indeed, K_1 is the one vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A,C}. - Emeric Deutsch, Jan 13 2014

Row sums of A235113.

LINKS

Table of n, a(n) for n=0..25.

F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, A closed formula for the number of convex permutominoes, arXiv:math/0702550 [math.CO], 2007.

Index entries for linear recurrences with constant coefficients, signature (8,-16).

FORMULA

a(n) = 8*a(n-1)-16*a(n-2), a(0) = 1, a(1) = 5. - Paul Barry, Mar 07 2003

G.f.: (1 - 3*x)/(1 - 4*x)^2. - Philippe Deléham, Dec 11 2008

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=0} 1/a(n) = 1024*log(4/3) - 880/3.

Sum_{n>=0} (-1)^n/a(n) = 688/3 - 1024*log(5/4). (End)

E.g.f.: exp(4*x)*(1 + x). - Stefano Spezia, Mar 05 2023

MATHEMATICA

LinearRecurrence[{8, -16}, {1, 5}, 22] (* Jean-François Alcover, Nov 06 2018 *)

PROG

(Sage) [lucas_number2(n, 4, 0)*n/2^10 for n in range(4, 26)] # Zerinvary Lajos, Mar 13 2009

CROSSREFS

Cf. A001792, A006234, A081105, A006234, A235113.

Cf. A002697, A034007, A079861, A045891, A087449.

KEYWORD

nonn,easy

AUTHOR

Benoit Cloitre, Feb 01 2003

EXTENSIONS

More terms from Stefano Spezia, Mar 05 2023

STATUS

approved

A081106

6th binomial transform of (1,1,0,0,0,0,...).

+10
5

1, 7, 48, 324, 2160, 14256, 93312, 606528, 3919104, 25194240, 161243136, 1027924992, 6530347008, 41358864384, 261213880320, 1645647446016, 10344069660672, 64885527871488, 406239826673664, 2538998916710400, 15843353240272896, 98716277881700352, 614234617930579968

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 5*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Index entries for linear recurrences with constant coefficients, signature (12,-36).

FORMULA

a(n) = 12*a(n-1) - 36*a(n-2) with n > 1, a(0) = 1, a(1) = 7.

a(n) = (n + 6)*6^(n-1).

G.f.: (1 - 5*x)/(1 - 6*x)^2.

E.g.f.: exp(6*x)*(1 + x). - Stefano Spezia, Mar 05 2023

MATHEMATICA

CoefficientList[Series[(1 - 5 x)/(1 - 6 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

LinearRecurrence[{12, -36}, {1, 7}, 30] (* Harvey P. Dale, Nov 07 2013 *)

PROG

(Magma) [(n+6)*6^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CROSSREFS

Cf. A079028, A081105, A006234.

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Mar 07 2003

STATUS

approved

A082308

Expansion of e.g.f. (1+x)*exp(4*x)*cosh(x).

+10
3

1, 5, 25, 127, 657, 3449, 18281, 97395, 519841, 2773741, 14776377, 78538343, 416367665, 2201517153, 11610231433, 61078202971, 320570884929, 1678897264085, 8775159682649, 45780628812879, 238431945108433

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Binomial transform of A082307.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (16,-94,240,-225).

FORMULA

a(n) = (A081105(n) + A006234(n))/2.

a(n) = ((n+3)*3^(n-1) + (n+5)*5^(n-1))/2.

G.f.: ((1-4*x)/(1-5*x)^2 + (1-2*x)/(1-3*x)^2)/2.

E.g.f.: (1+x)*exp(4*x)*cosh(x) = (1+x)*(exp(5*x) + exp(3*x))/2.

MATHEMATICA

With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[4*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)

PROG

(PARI) x='x+O('x^30); Vec(serlaplace((1+x)*exp(4*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(4*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018

CROSSREFS

Cf. A082309.

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 09 2003

STATUS

approved

A089944

Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

+10
2

1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 5, 20, 15, 5, 1, 6, 48, 54, 24, 6, 1, 7, 112, 189, 112, 35, 7, 1, 8, 256, 648, 512, 200, 48, 8, 1, 9, 576, 2187, 2304, 1125, 324, 63, 9, 1, 10, 1280, 7290, 10240, 6250, 2160, 490, 80, 10, 1, 11, 2816, 24057, 45056, 34375, 14256, 3773, 704

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The main diagonal is A089945: {T(n,n)=(2*n+1)*(n+1)^(n-1), n>=0}; the hyperbinomial transform of the main diagonal is the next lower diagonal in the array (A089946): {T(n+1,n) = 2*(n+1)*(n+2)^(n-1), n>=0}.

LINKS

Table of n, a(n) for n=0..62.

FORMULA

T(n,k) = (k+n+1)*(n+1)^(k-1).

E.g.f.: (1+x)*exp(x)/((1-y*exp(x)).

EXAMPLE

Rows begin:

{1, 2, 3, 4, 5, 6, 7,..},

{1, 3, 8, 20, 48, 112, 256,..},

{1, 4, 15, 54, 189, 648, 2187,..},

{1, 5, 24, 112, 512, 2304, 10240,..},

{1, 6, 35, 200, 1125, 6250, 34375,..},