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A176332 Row sums of triangle A176331. 6

%I

%S 1,2,5,16,56,202,741,2752,10318,38972,148070,565280,2166646,8332378,

%T 32136205,124249856,481433286,1868972828,7267804550,28304698336,

%U 110383060776,431000853028,1684754608210,6592277745536,25818887839956

%N Row sums of triangle A176331.

%C Hankel transform is A176333.

%C Let A(n) denote the n X n array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i-1 times to the tuple ((sqrt(-1))^m, 1 <= m <= n). Then the negative of the real part of the (n, n)-entry of A(n) equals a(n-2) for all n > 2. - _John M. Campbell_, Jan 20 2019

%H Vincenzo Librandi, <a href="/A176332/b176332.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(j,n-k)*binomial(j,k)*(-1)^(n-j).

%F Logarithm g.f.: arctan(x*C(x)) = Sum_{n>=1} a(n)/n*x^n, where C(x) = (1-sqrt(1-4*x))/(2*x) (A000108). - _Vladimir Kruchinin_, Aug 10 2010

%F Conjecture: 6*n*a(n) + 2*(-17*n+10)*a(n-1) + (47*n-60)*a(n-2) + 10*(-3*n+5)*a(n-3) + 4*(2*n-5)*a(n-4) = 0. - _R. J. Mathar_, Nov 24 2012

%F Recurrence: 2*n*(5*n-8)*a(n) = 2*(25*n^2 - 50*n + 18)*a(n-1) - (45*n^2 - 92*n + 36)*a(n-2) + 2*(2*n-3)*(5*n-3)*a(n-3). - _Vaclav Kotesovec_, Feb 12 2014

%F a(n) ~ 4^(n+1) / (5*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 12 2014

%p a:=n->add(add(binomial(j,n-k)*binomial(j,k)*(-1)^(n-j),j=0..n),k=0..n): seq(a(n),n=0..30); # _Muniru A Asiru_, Jan 23 2019

%t f[n_]:= (-1)^n*Sum[Binomial[n+k, k] Cos[Pi(n+k)/2], {k, 0, n}]; Array[f, 24, 0] (* _Robert G. Wilson v_, Apr 02 2012 *)

%o (PARI) {a(n) = sum(k=0,n, sum(j=0,n, (-1)^(n-j)*binomial(j,n-k)* binomial(j,k))) };vector(30, n, n--; a(n)) \\ _G. C. Greubel_, Feb 21 2019

%o (PARI) a(n) = {my(v = vector(n, k, I^k)); for (k=1, n-1, v = vector(n, i, sum(j=1, i, v[j]));); -real(v[n]);} \\ _Michel Marcus_, Feb 25 2019

%o (MAGMA) [(&+[ (&+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]]): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 21 2019

%o (Sage) [sum(sum((-1)^(n-j)*binomial(j,n-k)*binomial(j,k) for j in (0..n)) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Feb 21 2019

%o (GAP) List([0..30], n -> Sum([0..n], k -> Sum([0..n], j -> (-1)^(n-j)* Binomial(j, n-k)*Binomial(j, k) ))) # _G. C. Greubel_, Feb 22 2019

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 15 2010

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Last modified August 3 18:55 EDT 2021. Contains 346441 sequences. (Running on oeis4.)