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A154286 a(n) = E(k)C(n+k,k) = Euler(k)*Pascal(n+k,k) for k=4. 14

%I

%S 5,25,75,175,350,630,1050,1650,2475,3575,5005,6825,9100,11900,15300,

%T 19380,24225,29925,36575,44275,53130,63250,74750,87750,102375,118755,

%U 137025,157325,179800,204600,231880,261800,294525,330225,369075,411255

%N a(n) = E(k)C(n+k,k) = Euler(k)*Pascal(n+k,k) for k=4.

%C + a(n) = E(4)binom{n+4}{4} where E(n) are the Euler number in the enumeration A122045.

%C + a(n) is the special case k=4 in the sequence of diagonals in the triangle A153641.

%C + a(n) is the 5th row in A093375.

%C + a(n) is the 5th column in A103406.

%C + a(n) is the 5th antidiagonal in A103283.

%C + (a(n+1)-a(n))/5 are the pyramidal numbers A000292 (n>1).

%C + (a(n+2)-2a(n+1)+a(n))/5 are the triangular numbers A000217 (n>2).

%C + (a(n+3)-3a(n+2)+3a(n+1)-a(n))/5 are the natural numbers A000027 (n>3).

%C Number of orbits of Aut(Z^7) as function of the infinity norm (n+4) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 107520. - _Philippe A.J.G. Chevalier_, Dec 28 2015

%H G. C. Greubel, <a href="/A154286/b154286.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (n+1)*(n+2)*(n+3)*(n+4)*5/24.

%F a(n) = a(n-1)*(n+4)/n (n>0), a(0)=5.

%F O.g.f.: 5/(1-x)^5.

%F E.g.f.: (5/24)*x*(24 + 36*x + 12*x^2 + x^3)*exp(x). - _G. C. Greubel_, Sep 09 2016

%F a(n) = 5*A000332(n+4). - _Michel Marcus_, Sep 10 2016

%p seq(euler(4)*binomial(n+4,4),n=0..32);

%t CoefficientList[Series[-5/(x - 1)^5, {x, 0, 35}], x] (* _Robert G. Wilson v_, Jan 29 2015 *)

%t Table[(n + 1)*(n + 2)*(n + 3)*(n + 4)*5/24, {n, 0, 25}] (* _G. C. Greubel_, Sep 09 2016 *)

%o (MAGMA) [(n+1)*(n+2)*(n+3)*(n+4)*5 div 24: n in [0..40]]; // _Vincenzo Librandi_, Sep 10 2016

%o (PARI) x='x+O('x^99); Vec(5/(1-x)^5) \\ _Altug Alkan_, Sep 10 2016

%Y Cf. A000217, A153641, A000579.

%K easy,nonn

%O 0,1

%A _Peter Luschny_, Jan 06 2009

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Last modified February 19 16:02 EST 2019. Contains 320311 sequences. (Running on oeis4.)