A118436 -id:A118436 - cp's OEIS frontend

A118435 Triangle T, read by rows, equal to the matrix product T = H[C^-1]H, where H is the self-inverse triangle A118433 and C is Pascal's triangle.

1, 1, 1, -3, 2, 1, -11, 15, 3, 1, 25, -44, -18, 4, 1, 41, -115, -110, 50, 5, 1, -43, 246, 375, -220, -45, 6, 1, 29, 315, 861, -805, -385, 105, 7, 1, -335, 232, -1204, 2296, 1750, -616, -84, 8, 1, -1199, 3033, 1044, 3780, 5166, -2898, -924, 180, 9, 1

Offset: 0

Paul D. Hanna, Apr 28 2006

The matrix inverse of H*[C^-1]*H is H*C*H = A118438, where H^2 = I (identity). The matrix log, log(T) = A118441, is a matrix square root of a triangular matrix with a single diagonal (two rows down from the main diagonal).

			Triangle begins:
  1;
  1, 1;
  -3, 2, 1;
  -11, 15, 3, 1;
  25,-44,-18, 4, 1;
  41,-115,-110, 50, 5, 1;
  -43, 246, 375,-220,-45, 6, 1;
  29, 315, 861,-805,-385, 105, 7, 1;
  -335, 232,-1204, 2296, 1750,-616,-84, 8, 1;
  -1199, 3033, 1044, 3780, 5166,-2898,-924, 180, 9, 1;
  ...
The matrix log, log(T) = A118441, starts:
  0;
  1, 0;
  -4, 2, 0;
  -12, 12, 3, 0;
  32,-48,-24, 4, 0;
  80,-160,-120, 40, 5, 0;
  ...
where matrix square, log(T)^2, is a single diagonal:
  0;
  0,0;
  2,0,0;
  0,6,0,0;
  0,0,12,0,0;
  0,0,0,20,0,0;
  ...

Cf. A118436 (column 0), A118437 (row sums), A118438 (matrix inverse), A118441 (matrix log), A118433 (self-inverse H).

nmax = 12;
h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k);
H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}];
Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}];
Tn = H.Inverse[Cn].H;
T[n_, k_] := Tn[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2024 *)

{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1)*(-1)^(r\2- (c-1)\2+r-c))),C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))));(M*C^-1*M)[n+1,k+1]}

Since T + T^-1 = C + C^-1, then [T^-1](n,k) = (1+(-1)^(n-k))*C(n,k) - T(n,k) is a formula for the matrix inverse T^-1 = A118438.

A118437 Row sums of triangle A118435.

Original entry on oeis.org

1, 2, 0, 8, -32, -128, 320, 128, 2048, 8192, -15360, 2048, -131072, -524288, 1064960, 32768, 8388608, 33554432, -66846720, 524288, -536870912, -2147483648, 4299161600, 8388608, 34359738368, 137438953472, -274810798080, 134217728

Offset: 0

Paul D. Hanna, Apr 28 2006

sign

Index entries for linear recurrences with constant coefficients, signature (2, -4, 8, -64, 128, -256, 512).

Cf. A118435 (triangle), A118436 (column 0).

nmax = 27;
h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k);
H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}];
Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}];
Total /@ (H.Inverse[Cn].H ) (* Jean-François Alcover, Apr 08 2024 *)

{a(n)=polcoeff((1+8*x^3-32*x^5+384*x^6-256*x^7)/(1-2*x)/(1+4*x^2)/(1+64*x^4+x*O(x^n)),n)}

G.f.: (1+8*x^3-32*x^5+384*x^6-256*x^7)/(1-2*x)/(1+4*x^2)/(1+64*x^4).

A118439 Column 0 of triangle A118438.

Original entry on oeis.org

1, -1, 5, 11, -23, -41, 45, -29, 337, 1199, -3115, -6469, 10297, 8839, 16125, 108691, -354143, -873121, 1721765, 2521451, -1476983, 6699319, -34182195, -103232189, 242017777, 451910159, -597551755, -130656229, -2465133863, -10513816601, 29729597085, 66349305331, -116749235903

Offset: 0

Paul D. Hanna, Apr 28 2006

sign

Index entries for linear recurrences with constant coefficients, signature (0, -5, 0, -19, 0, 25).

Cf. A118438 (triangle), A118440 (row sums); A118436 (column 0 of inverse).

LinearRecurrence[{0, -5, 0, -19, 0, 25}, {1, -1, 5, 11, -23, -41}, 33] (* Jean-François Alcover, Apr 08 2024 *)

{a(n)=polcoeff((1-x+10*x^2+6*x^3+21*x^4-5*x^5)/(1-x^2)/(1+6*x^2+25*x^4+x*O(x^n)),n)}

G.f.: (1-x+10*x^2+6*x^3+21*x^4-5*x^5)/(1-x^2)/(1+6*x^2+25*x^4).

cp's OEIS Frontend

A118435 Triangle T, read by rows, equal to the matrix product T = H[C^-1]H, where H is the self-inverse triangle A118433 and C is Pascal's triangle.

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A118437 Row sums of triangle A118435.

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A118439 Column 0 of triangle A118438.

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cp's OEIS Frontend

A118435 Triangle T, read by rows, equal to the matrix product T = H*[C^-1]*H, where H is the self-inverse triangle A118433 and C is Pascal's triangle.

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PARI

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A118437 Row sums of triangle A118435.

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Programs

Mathematica

PARI

Formula

A118439 Column 0 of triangle A118438.

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Mathematica

PARI

Formula

A118435 Triangle T, read by rows, equal to the matrix product T = H[C^-1]H, where H is the self-inverse triangle A118433 and C is Pascal's triangle.