A138188 -id:A138188 - cp's OEIS frontend

A138187 Hankel transform of binomial(2*n+3, n).

1, -4, 3, 3, -8, 5, 5, -12, 7, 7, -16, 9, 9, -20, 11, 11, -24, 13, 13, -28, 15, 15, -32, 17, 17, -36, 19, 19, -40, 21, 21, -44, 23, 23, -48, 25, 25, -52, 27, 27, -56, 29, 29, -60, 31, 31, -64, 33, 33, -68, 35, 35, -72, 37, 37, -76, 39, 39, -80, 41

Offset: 0

Paul Barry, Mar 04 2008

Hankel transform of A002054(n+1).
Hankel transform of A002054(n) is A057078(n+1).
Partial sums are A138188.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-3,-2,-1).

Cf. A002054, A057078, A138188.

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1-2*x-2*x^2-x^3)/(1+x+x^2)^2 )); // G. C. Greubel, Jun 16 2021

a[n_]:= a[n]= Sum[(-1)^(n-k+1)*(n+k+2)*Binomial[n+k+1, 2*k], {k, 0, n+1}];
Table[a[n], {n, 0, 65}] (* G. C. Greubel, Jun 16 2021 *)

@CachedFunction
def A138187(n):
    if (n%3==0): return 2*(n//3) +1
    elif (n%3==1): return -4*((n//3) +1)
    else: return 2*(n//3) +3
[A138187(n) for n in (0..65)] # G. C. Greubel, Jun 16 2021

G.f.: (1 -2*x -2*x^2 -x^3)/(1 +x +x^2)^2.
a(n) = Sum_{k=0..n} (-1)^(n-k+1)*(n+k+2)*binomial(n+k+1, 2*k). - Paul Barry, Apr 19 2010
a(n) = 2*floor(n/3) + 1 if (n mod 3) = 0, -4*(floor(n/3) + 1) if (n mod 3) = 1 and 2*floor(n/3) + 3 if (n mod 3) = 2. - G. C. Greubel, Jun 16 2021

A138189 Sequence built on pattern (1,-even,-even) beginning 1,-2,-2.

Original entry on oeis.org

1, -2, -2, 1, -4, -4, 1, -6, -6, 1, -8, -8, 1, -10, -10, 1, -12, -12, 1, -14, -14, 1, -16, -16, 1, -18, -18, 1, -20, -20, 1, -22, -22, 1, -24, -24, 1, -26, -26, 1, -28, -28, 1, -30, -30, 1, -32, -32, 1, -34, -34, 1, -36, -36, 1, -38, -38, 1, -40, -40

Offset: 0

Paul Barry, Mar 04 2008

Partial sums of A138188.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A110090, A138188.

b:= [n le 2 select 0 else Abs(2*Self(n-1) -Self(n-2)) -Self(n-1)-1: n in [1..120]];
A138189:= func< n | -b[n+3] >;
[A138189(n): n in [0..100]]; // G. C. Greubel, Jun 16 2021

Join[{1},Riffle[Flatten[{-2#,-2#}&/@Range[25]],1,3]] (* Harvey P. Dale, Nov 02 2011 *)

@CachedFunction
def A138189(n):
    if (n%3==0): return 1
    elif (n%3==1): return -2*(n//3 +1)
    else: return -2*(n//3 +1)
[A138189(n) for n in (0..100)] # G. C. Greubel, Jun 16 2021

G.f.: (1 -2*x -2*x^2 -x^3)/(1 -2*x^3 +x^6).
From G. C. Greubel, Jun 16 2021: (Start)
a(n) = -b(n+3), where b(n) = abs(2*b(n-1) - b(n-2)) - b(n-1) - 1 and b(1) = b(2) = 0.
a(n) = 1 if (n mod 3) = 0, -2*(floor(n/3) + 1) if (n mod 3) = 1 or (n mod 3) = 2. (End)

cp's OEIS Frontend

A138187 Hankel transform of binomial(2*n+3, n).

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