A115713 -id:A115713 - cp's OEIS frontend

A115717 A divide-and-conquer triangle related to A007583.

1, 0, 1, 3, -1, 1, 0, 0, 0, 1, 0, 4, -1, -1, 1, 0, 0, 0, 0, 0, 1, 12, -4, 4, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 16, -4, -4, 4, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 48, -16, 16, 0, -4, -4, 4, 0, 0, 0, 0, 0, -1, -1, 1

Offset: 0

Paul Barry, Jan 29 2006

Product of (1-x, x), which is A167374, and number triangle A115715.

			Triangle begins
   1;
   0,   1;
   3,  -1,  1;
   0,   0,  0,  1;
   0,   4, -1, -1,  1;
   0,   0,  0,  0,  0,  1;
  12,  -4,  4,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  4,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,  16, -4, -4,  4,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  0,  0,  4,  0,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  48, -16, 16,  0, -4, -4,  4,  0,  0,  0,  0,  0, -1, -1,  1;

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

Cf. A007583, A115715, A115716 (row sums), A167374.

A115717 := proc(n,k)
    add( A167374(n,j)*A115715(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 07 2016

A167374[n_, k_]:= If[k>n-2, (-1)^(n-k), 0];
g[n_, k_]:= g[n, k]= If[k==n, 1, If[k==n-1, -Mod[n, 2], If[n==2*k+2, -4, 0]]]; (* g = A115713 *)
f[n_, k_]:= f[n, k]= If[k==n, 1, -Sum[f[n,j]*g[j,k], {j,k+1,n}]]; (* f=A115715 *)
A115717[n_, k_]:= A115717[n, k]= Sum[A167374[n,j]*f[j,k], {j,k,n}];
Table[A115717[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

@cached_function
def A115717(n,k):
    def A167374(n, k):
        if (k>n-2): return (-1)^(n-k)
        else: return 0
    def A115713(n,k):
        if (k==n): return 1
        elif (k==n-1): return -(n%2)
        elif (n==2*k+2): return -4
        else: return 0
    def A115715(n,k):
        if (k==0): return 4^(floor(log(n+2, 2)) -1)
        elif (k==n): return 1
        elif (k==n-1): return (n%2)
        else: return (-1)*sum( A115715(n,j+k+1)*A115713(j+k+1,k) for j in (0..n-k-1) )
    return sum( A167374(n, j+k)*A115715(j+k, k) for j in (0..n-k) )
flatten([[A115717(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 23 2021

Sum_{k=0..n} T(n, k) = A115716(n).

T(n ,k) = Sum_{j=k..n} A167374(n, j)*A115715(j, k). - R. J. Mathar, Sep 07 2016

A115715 A divide-and-conquer triangle.

Original entry on oeis.org

1, 1, 1, 4, 0, 1, 4, 0, 1, 1, 4, 4, 0, 0, 1, 4, 4, 0, 0, 1, 1, 16, 0, 4, 0, 0, 0, 1, 16, 0, 4, 0, 0, 0, 1, 1, 16, 0, 4, 4, 0, 0, 0, 0, 1, 16, 0, 4, 4, 0, 0, 0, 0, 1, 1, 16, 16, 0, 0, 4, 0, 0, 0, 0, 0, 1, 16, 16, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 16, 16, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 1, 16, 16, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Paul Barry, Jan 29 2006

			Triangle begins
   1;
   1,  1;
   4,  0,  1;
   4,  0,  1,  1;
   4,  4,  0,  0,  1;
   4,  4,  0,  0,  1,  1;
  16,  0,  4,  0,  0,  0,  1;
  16,  0,  4,  0,  0,  0,  1,  1;
  16,  0,  4,  4,  0,  0,  0,  0,  1;
  16,  0,  4,  4,  0,  0,  0,  0,  1,  1;
  16, 16,  0,  0,  4,  0,  0,  0,  0,  0,  1;
  16, 16,  0,  0,  4,  0,  0,  0,  0,  0,  1,  1;
  16, 16,  0,  0,  4,  4,  0,  0,  0,  0,  0,  0,  1;
  16, 16,  0,  0,  4,  4,  0,  0,  0,  0,  0,  0,  1,  1;
  64,  0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1;
  64,  0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1,  1;
  64,  0, 16,  0,  0,  0,  4,  4,  0,  0,  0,  0,  0,  0,  0,  0,  1;

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

Cf. A032925 (row sums), A115639 (first column), A115713 (inverse).

A115715 := proc(n,k)
    option remember;
    if n = k then
        1;
    elif k > n then
        0;
    else
        -add(procname(n,l)*A115713(l,k),l=k+1..n) ;
    end if;
end proc:
seq(seq(A115715(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Sep 07 2016

A115713[n_, k_]:= If[k==n, 1, If[k==n-1, ((-1)^n-1)/2, If[n==2*k+2, -4, 0]]];
T[n_, k_]:= T[n, k]= If[k==n, 1, -Sum[T[n, j]*A115713[j, k], {j, k+1, n}]];
Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

@CachedFunction
def A115713(n,k):
    if (k==n): return 1
    elif (k==n-1): return -(n%2)
    elif (n==2*k+2): return -4
    else: return 0
def A115715(n,k):
    if (k==0): return 4^(floor(log(n+2, 2)) -1)
    elif (k==n): return 1
    elif (k==n-1): return (n%2)
    else: return (-1)*sum( A115715(n,j)*A115713(j,k) for j in (k+1..n) )
flatten([[A115715(n,k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021

Sum_{=0..n} T(n, k) = A032925(n).
T(n, 0) = A115639(n).
T(n, k) = 1 if n = k, otherwise T(n, k) = (-1)*Sum_{j=k+1..n} T(n, j)*A115713(j, k). - R. J. Mathar, Sep 07 2016

A115714 Periodic {1,-1,-3,0,1,-5,1,0,-3,-1,1,-4}.

Original entry on oeis.org

1, -1, -3, 0, 1, -5, 1, 0, -3, -1, 1, -4, 1, -1, -3, 0, 1, -5, 1, 0, -3, -1, 1, -4, 1, -1, -3, 0, 1, -5, 1, 0, -3, -1, 1, -4, 1, -1, -3, 0, 1, -5, 1, 0, -3, -1, 1, -4, 1, -1, -3, 0, 1, -5, 1, 0, -3, -1, 1, -4, 1, -1, -3, 0, 1, -5, 1, 0, -3, -1, 1, -4

Offset: 0

Paul Barry, Jan 29 2006

Diagonal sums of number triangle A115713.

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

Cf. A115713.

LinearRecurrence[{-1,-1,0,1,1,1}, {1,-1,-3,0,1,-5}, 80] (* G. C. Greubel, Nov 23 2021 *)

def A115714_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x^2+4*x^3+3*x^4+4*x^5)/(1+x+x^2-x^4-x^5-x^6) ).list()
A115714_list(80) # G. C. Greubel, Nov 23 2021

G.f.: (1 - 3*x^2 + 4*x^3 + 3*x^4 + 4*x^5)/(1 + x + x^2 - x^4 - x^5 - x^6).

a(n) = Sum_{k=0..floor(n/2)} A115713(n-k, k).

A115718 Inverse of number triangle A115717; a divide-and-conquer related triangle.

Original entry on oeis.org

1, 0, 1, -3, 1, 1, 0, 0, 0, 1, -3, -3, 1, 1, 1, 0, 0, 0, 0, 0, 1, -3, -3, -3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, -3, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -3, -3, -3, -3, -3, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Jan 29 2006

Product of A115713 and (1/(1-x), x).
Row sums are 1,1,-1,1,-3,1,-5,1,-7,1, ... with g.f. (1+x-3*x^2-x^3)/(1-x^2)^2.
Row sums of inverse are A115716.

			Triangle begins
   1;
   0,  1;
  -3,  1,  1;
   0,  0,  0,  1;
  -3, -3,  1,  1,  1;
   0,  0,  0,  0,  0,  1;
  -3, -3, -3,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1;
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  -3, -3, -3, -3, -3, -3, -3,  1,  1,  1,  1,  1,  1,  1,  1;

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

Cf. A115713, A115716, A115717, A237420.

T[n_, k_]:= If[OddQ[n], If[kG. C. Greubel, Nov 29 2021 *)

def A115718(n,k):
    if (n%2==0): return 0 if (kA115718(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 29 2021

From G. C. Greubel, Nov 29 2021: (Start)
T(2*n, k) = -3 if (k < n/2) otherwise 1.
T(2*n+1, k) = 0 if (k < n) otherwise 1.
Sum_{k=0..n} T(n, k) = (1/2)*(2 + (1 + (-1)^n)*n) = 1 + A237420(n). (End)

cp's OEIS Frontend

A115717 A divide-and-conquer triangle related to A007583.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A115715 A divide-and-conquer triangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A115714 Periodic {1,-1,-3,0,1,-5,1,0,-3,-1,1,-4}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A115718 Inverse of number triangle A115717; a divide-and-conquer related triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula