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%I #39 Oct 01 2023 12:17:37
%S 1,20,240,2240,17920,129024,860160,5406720,32440320,187432960,
%T 1049624576,5725224960,30534533120,159719096320,821412495360,
%U 4161823309824,20809116549120,102821517066240,502682972323840,2434043865989120,11683410556747776,55635288365465600
%N 5-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^5.
%C Also convolution of A020920 with A000984 (central binomial coefficients).
%C With a different offset, number of n-permutations (n=5) of 5 objects u, v, w, z, x with repetition allowed, containing exactly four (4)u's. Example: a(1)=20 because we have uuuuv, uuuvu, uuvuu, uvuuu, vuuuu, uuuuw, uuuwu, uuwuu, uwuuu, wuuuu, uuuuz, uuuzu, uuzuu, uzuuu, zuuuu, uuuux, uuuxu, uuxuu, uxuuu and xuuuu. - _Zerinvary Lajos_, May 19 2008
%C Also convolution of A000302 with A038846, also convolution of A002457 with A020918, also convolution of A002697 with A038845, also convolution of A002802 with A002802. [_Rui Duarte_, Oct 08 2011]
%H Vincenzo Librandi, <a href="/A040075/b040075.txt">Table of n, a(n) for n = 0..400</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20, -160, 640, -1280, 1024).
%F a(n) = binomial(n+4, 4)*4^n.
%F G.f.: 1/(1-4*x)^5.
%F a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10 = 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) with f(k)=A000984(k). - _Rui Duarte_, Oct 08 2011
%F E.g.f.: (3 + 48*x + 144*x^2 + 128*x^3 + 32*x^4)*exp(4*x)/3. - _G. C. Greubel_, Jul 20 2019
%F From _Amiram Eldar_, Mar 25 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 376/3 - 432*log(4/3).
%F Sum_{n>=0} (-1)^n/a(n) = 2000*log(5/4) - 1336/3. (End)
%p seq(seq(binomial(i, j)*4^(i-4), j =i-4), i=4..22); # _Zerinvary Lajos_, Dec 03 2007
%p seq(binomial(n+4,4)*4^n,n=0..30); # _Zerinvary Lajos_, May 19 2008
%p spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, Z, Z, Z, B, B, B, B)}, labeled]: seq(combstruct[count](spec, size=n)/24, n=4..34); # _Zerinvary Lajos_, Apr 05 2009
%t Table[Binomial[n+4,4]*4^n, {n,0,30}] (* _Michael De Vlieger_, Aug 21 2015 *)
%o (Sage) [lucas_number2(n, 4, 0)*binomial(n,4)/2^8 for n in range(4, 34)] # _Zerinvary Lajos_, Mar 11 2009
%o (Magma) [4^n*Binomial(n+4, 4): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%o (PARI) vector(30, n, n--; 4^n*binomial(n+4, 4)) \\ _G. C. Greubel_, Jul 20 2019
%o (GAP) List([0..30], n-> 4^n*Binomial(n+4, 4)); # _G. C. Greubel_, Jul 20 2019
%Y Cf. A000302, A020920, A000984.
%Y Cf. A038231.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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