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A038845 3-fold convolution of A000302 (powers of 4). 24
1, 12, 96, 640, 3840, 21504, 114688, 589824, 2949120, 14417920, 69206016, 327155712, 1526726656, 7046430720, 32212254720, 146028888064, 657129996288, 2937757630464, 13056700579840, 57724360458240, 253987186016256 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also convolution of A002802 with A000984 (central binomial coefficients)

With a different offset, number of n-permutations of 5 objects u, v, w, z, x with repetition allowed, containing exactly two u's. - Zerinvary Lajos, Dec 29 2007

Also convolution of A000302 with A002697, also convolution of A002457 with itself. - Rui Duarte, Oct 08 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

FORMULA

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

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

a(n) = Sum_{a+b+c+d+e+f=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004

a(n) = binomial(n+2,n) * 4^n. - Rui Duarte, Oct 08 2011

E.g.f.: (1 + 8*x + 8*x^2)*exp(4*x). - G. C. Greubel, Jul 20 2019

MAPLE

seq((n+2)*(n+1)*4^n/2, n=0..30); # Zerinvary Lajos, Apr 25 2007

seq(seq(binomial(i+2, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007

seq(seq(binomial(i+2, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 29 2007

MATHEMATICA

Table[4^n*Binomial[n+2, n], {n, 0, 30}] (* G. C. Greubel, Jul 20 2019 *)

PROG

(Sage) [lucas_number2(n, 4, 0)*binomial(n, 2)/2^4 for n in range(2, 30)] # Zerinvary Lajos, Mar 11 2009

(MAGMA) [4^n*Binomial(n+2, 2): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

(PARI) a(n)=(n+2)*(n+1)<<(2*n-1) \\ Charles R Greathouse IV, Aug 21 2015

(GAP) List([0..30], n-> 4^n*Binomial(n+2, n) ); # G. C. Greubel, Jul 20 2019

CROSSREFS

Cf. A000302, A002802, A000984, A052780, A038231.

Sequence in context: A138162 A264418 A073392 * A204623 A270568 A223151

Adjacent sequences:  A038842 A038843 A038844 * A038846 A038847 A038848

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified December 1 16:17 EST 2020. Contains 338844 sequences. (Running on oeis4.)