A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

1, 24, 336, 3584, 32256, 258048, 1892352, 12976128, 84344832, 524812288, 3148873728, 18320719872, 103817412608, 574988746752, 3121367482368, 16647293239296, 87398289506304, 452414675091456, 2312341672689664, 11683410556747776, 58417052783738880, 289303499500421120

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).

With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - Zerinvary Lajos, Jun 16 2008

Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - 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 (24,-240,1280,-3840,6144,-4096).

Crossrefs

Cf. A038231.

Programs

GAP

List([0..30], n-> 4^n*Binomial(n+5,5)); # G. C. Greubel, Jul 20 2019

Magma

[4^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+5,5)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

Mathematica

CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* Harvey P. Dale, Mar 24 2018 *)

PARI

Vec(1/(1-4*x)^6 + O(x^30)) \\ Michel Marcus, Aug 21 2015

Sage

[lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # Zerinvary Lajos, Mar 11 2009

Formula

a(n) = binomial(n+5, 5)*4^n.

G.f.: 1/(1-4*x)^6.

a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - Rui Duarte, Oct 08 2011

E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019

From Amiram Eldar, Mar 25 2022: (Start)

Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.

Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)