A124749 -id:A124749 - cp's OEIS frontend

A124752 Inverse of number triangle A124749.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 3, 3, 3, 3, 2, 1, 0, 0, 0, 4, 4, 4, 4, 3, 2, 1, 0, 0, 0, 9, 9, 9, 9, 7, 5, 3, 1

Offset: 0

Paul Barry, Nov 06 2006

Row sums (and fourth column) are A124753.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 0, 0, 1,
0, 0, 0, 1, 1,
0, 0, 0, 1, 1, 1,
0, 0, 0, 1, 1, 1, 1,
0, 0, 0, 2, 2, 2, 2, 1,
0, 0, 0, 3, 3, 3, 3, 2, 1,
0, 0, 0, 4, 4, 4, 4, 3, 2, 1,
0, 0, 0, 9, 9, 9, 9, 7, 5, 3, 1

Cf. A124747.

A124750 Expansion of (1 + x + x^2)/(1 - x^3 + x^4).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 0, -1, 0, 0, -1, 1, 0, -1, 2, -1, -1, 3, -3, 0, 4, -6, 3, 4, -10, 9, 1, -14, 19, -8, -15, 33, -27, -7, 48, -60, 20, 55, -108, 80, 35, -163, 188, -45, -198, 351, -233, -153, 549, -584, 80, 702, -1133, 664, 622, -1835, 1797, -42, -2457, 3632, -1839

Offset: 0

Paul Barry, Nov 06 2006

Row sums of number triangle A124749.

Let A(n) denote the n X n matrix with 1's along and everywhere above the main diagonal, 1's along the sub-sub-subdiagonal, and 0's everywhere else; for n>3, a(n) equals (-1)^(n+1) times the sum of the coefficients of the characteristic polynomial of A(n-3) (see Mathematica code below). - John M. Campbell, Mar 10 2012

Index entries for linear recurrences with constant coefficients, signature (0,0,1,-1).

Cf. A124749.

A[n_] := Array[Sum[KroneckerDelta[#1, #2 - j], {j, 0, n}] + KroneckerDelta[#1, #2 + 3] &, {n, n}]; Table[(-1)^(r + 1)*Total[CoefficientList[CharacteristicPolynomial[A[r - 3], x],  x]], {r, 4, 60}] (* John M. Campbell, Mar 10 2012 *)
CoefficientList[Series[(1+x+x^2)/(1-x^3+x^4),{x,0,70}],x] (* or *) LinearRecurrence[{0,0,1,-1},{1,1,1,1},70] (* Harvey P. Dale, Jun 06 2018 *)

G.f.: (1 + x + x^2)/(1 - x^3 + x^4).
a(n) = Sum_{k=0..n} binomial(floor(k/3), n-k) * (-1)^(n-k).
a(n) = a(n-3) - a(n-4). - Wesley Ivan Hurt, May 02 2021

A124751 Expansion of (1+x^2+x^4)/(1-x^6+x^7).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, -1, 1, -1, 1, -1, 1, -2, 2, -2, 2, -2, 2, -3, 4, -4, 4, -4, 4, -5, 7, -8, 8, -8, 8, -9, 12, -15, 16, -16, 16, -17, 21, -27, 31, -32, 32, -33, 38, -48, 58, -63, 64, -65, 71, -86, 106, -121, 127, -129, 136, -157, 192, -227, 248

Offset: 0

Paul Barry, Nov 06 2006

Diagonal sums of number triangle A124749.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,-1).

CoefficientList[Series[(1+x^2+x^4)/(1-x^6+x^7),{x,0,100}],x] (* or *) LinearRecurrence[{0,0,0,0,0,1,-1},{1,0,1,0,1,0,1},100] (* Harvey P. Dale, Mar 10 2017 *)

a(n)=sum{k=0..floor(n/2), C(floor(k/3),n-2k)*(-1)^n}

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A124752 Inverse of number triangle A124749.

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A124750 Expansion of (1 + x + x^2)/(1 - x^3 + x^4).

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A124751 Expansion of (1+x^2+x^4)/(1-x^6+x^7).

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