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URL: https://oeis.org/A176334

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A176334
Diagonal sums of number triangle A176331.
3
1, 1, 2, 4, 9, 21, 51, 124, 305, 755, 1879, 4698, 11792, 29694, 74984, 189811, 481498, 1223713, 3115200, 7942134, 20275362, 51823246, 132604193, 339644739, 870745187, 2234208932, 5737129623, 14742751524, 37909928908, 97543380598
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j,n-2k)*C(j,k)*(-1)^(n-k-j).
a(n) ~ phi^(2*n+3) / (4*5^(1/4)*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 08 2024
MAPLE
A176334 := proc(n)
add(add(binomial(j, n-2*k)*binomial(j, k)*(-1)^(n-k-j), j=0..n-k), k=0..floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015
MATHEMATICA
T[n_, k_]:= T[n, k]= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j, 0, n}]; Table[Sum[T[n-k, k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Dec 07 2019 *)
PROG
(PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(30, n, sum(j=0, (n-1)\2, T(n-j-1, j)) ) \\ G. C. Greubel, Dec 07 2019
(Magma) T:= func< n, k | &+[(-1)^(n-j)*Binomial(j, n-k)*Binomial(j, k): j in [0..n]] >;
[(&+[T(n-k, k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Dec 07 2019
(SageMath)
@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Dec 07 2019
(GAP)
T:= function(n, k)
return Sum([0..n], j-> (-1)^(n-j)*Binomial(j, k)*Binomial(j, n-k) );
end;
List([0..30], n-> Sum([0..Int(n/2)], j-> T(n-j, j) )); # G. C. Greubel, Dec 07 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 15 2010
STATUS
approved