A177229 -id:A177229 - cp's OEIS frontend

A177227 Triangle, read by rows, T(n, k) = -binomial(n,k) for 0 < k < n, otherwise T(n, k) = 2.

2, 2, 2, 2, -2, 2, 2, -3, -3, 2, 2, -4, -6, -4, 2, 2, -5, -10, -10, -5, 2, 2, -6, -15, -20, -15, -6, 2, 2, -7, -21, -35, -35, -21, -7, 2, 2, -8, -28, -56, -70, -56, -28, -8, 2, 2, -9, -36, -84, -126, -126, -84, -36, -9, 2, 2, -10, -45, -120, -210, -252, -210, -120, -45, -10, 2

Offset: 0

Roger L. Bagula, May 05 2010

This triangle may also be constructed in the following way. Let f_{n}(t) = d^n/dt^n t/(1+t) = (-1)^(n+1)*n!*(1+t)^(-n-1). Then the triangle is given as f_{n}(t)/((1+t)*f_{k}(t)*f_{n-k}(t)) when t = 1/2 (this sequence), t = 1/3 (A177228), and t = 1/4 (A177229).

			Triangle begins as:
  2;
  2,   2;
  2,  -2,   2;
  2,  -3,  -3,    2;
  2,  -4,  -6,   -4,    2;
  2,  -5, -10,  -10,   -5,    2;
  2,  -6, -15,  -20,  -15,   -6,    2;
  2,  -7, -21,  -35,  -35,  -21,   -7,    2;
  2,  -8, -28,  -56,  -70,  -56,  -28,   -8,   2;
  2,  -9, -36,  -84, -126, -126,  -84,  -36,  -9,   2;
  2, -10, -45, -120, -210, -252, -210, -120, -45, -10,  2;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A131130 (related to row sums), A177228, A177229.

A177227:= func< n,k | k eq 0 or k eq n select 2 else -Binomial(n,k) >;
[A177227(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2024

T[n_, k_]:= If[k==0 || k==n, 2, -Binomial[n,k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A177227(n,k): return 2 if (k==0 or k==n) else -binomial(n,k)
flatten([[A177227(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 09 2024

T(n, 0) = T(n, n) = 2, otherwise T(n, k) = -binomial(n,k).
Sum_{k=0..n} T(n, k) = -A131130(n-2) - 3*[n=0], n >= 1 (row sums).
From G. C. Greubel, Apr 09 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = 3*(1 + (-1)^n) - 4*[n=0].
Sum_{k=0..floor(n/2)} T(n-k,k) = (3/2)*(3 + (-1)^n - 2*[n=0])-Fibonacci(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 3*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)

Edited by G. C. Greubel, Apr 09 2024

A177228 Triangle read by rows: T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 3.

Original entry on oeis.org

3, 3, 3, 3, -2, 3, 3, -3, -3, 3, 3, -4, -6, -4, 3, 3, -5, -10, -10, -5, 3, 3, -6, -15, -20, -15, -6, 3, 3, -7, -21, -35, -35, -21, -7, 3, 3, -8, -28, -56, -70, -56, -28, -8, 3, 3, -9, -36, -84, -126, -126, -84, -36, -9, 3, 3, -10, -45, -120, -210, -252, -210, -120, -45, -10

Offset: 0

Roger L. Bagula, May 05 2010

This triangle may also be constructed in the following way. Let f_{n}(t) = d^n/dt^n t/(1+t) = (-1)^(n+1)*n!*(1+t)^(-n-1). Then the triangle is given as f_{n}(t)/((1+t)*f_{k}(t)*f_{n-k}(t)) when t = 1/2 (A177227), t = 1/3 (this sequence), and t = 1/4 (A177229).

This is the Pascal triangle A007318, with all entries sign-flipped, and 3's inserted at the beginning and end of each row. - R. J. Mathar, Mar 27 2024

			Triangle begins:
  3;
  3,   3;
  3,  -2,   3;
  3,  -3,  -3,    3;
  3,  -4,  -6,   -4,    3;
  3,  -5, -10,  -10,   -5,    3;
  3,  -6, -15,  -20,  -15,   -6,    3;
  3,  -7, -21,  -35,  -35,  -21,   -7,    3;
  3,  -8, -28,  -56,  -70,  -56,  -28,   -8,   3;
  3,  -9, -36,  -84, -126, -126,  -84,  -36,  -9,   3;
  3, -10, -45, -120, -210, -252, -210, -120, -45, -10,  3;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A177227, A177229.

A177228:= func< n,k | k eq 0 or k eq n select 3 else -Binomial(n,k) >;
[A177228(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2024

f := proc(n,t)
    if n = 0 then
        t/(1+t) ;
    else
        diff( t/(1+t),t$n) ;
        factor(%) ;
    end if;
end proc:
A177228 := proc(n,m)
    f(n,t)/f(m,t)/f(n-m,t) ;
    %/(1+t) ;
    subs(t=1/3,%) ;
end proc:
seq(seq( A177228(n,m),m=0..n),n=0..12) ; # R. J. Mathar, Mar 27 2024

T[n_, k_]:= If[k==0 || k==n, 3, -Binomial[n,k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A177228(n,k): return 3 if (k==0 or k==n) else -binomial(n,k)
flatten([[A177228(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 09 2024

T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 3.
Sum_{k=0..n} T(n, k) = 8 - 2^n, for n >= 1.
From G. C. Greubel, Apr 09 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = 4*(1 + (-1)^n) - 5*[n=0].
Sum_{k=0..floor(n/2)} T(n-k,k) = 2*(3+(-1)^n-2*[n=0])-Fibonacci(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 4*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)

Edited by G. C. Greubel, Apr 09 2024

cp's OEIS Frontend

A177227 Triangle, read by rows, T(n, k) = -binomial(n,k) for 0 < k < n, otherwise T(n, k) = 2.

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