A110324 -id:A110324 - cp's OEIS frontend

A110326 Diagonal sums of triangle A110324.

1, -1, -3, -2, -11, -3, -23, -4, -39, -5, -59, -6, -83, -7, -111, -8, -143, -9, -179, -10, -219, -11, -263, -12, -311, -13, -363, -14, -419, -15, -479, -16, -543, -17, -611, -18, -683, -19, -759, -20, -839, -21, -923, -22, -1011, -23, -1103, -24, -1199, -25, -1299, -26, -1403, -27, -1511, -28, -1623, -29, -1739

Offset: 0

Paul Barry, Jul 20 2005

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

G.f.: (1-x-6x^2+x^3+x^4)/((1+x)^3(1-x)^3); a(n)=3a(n-2)-3a(n-4)+a(n-6); a(n)=-(n^2+3n-1)/4-(n^2+n-3)(-1)^n/4.

A110321 A Jacobsthal number related number triangle.

Original entry on oeis.org

1, 1, 1, 6, 2, 1, 30, 18, 3, 1, 264, 120, 36, 4, 1, 2520, 1320, 300, 60, 5, 1, 30960, 15120, 3960, 600, 90, 6, 1, 428400, 216720, 52920, 9240, 1050, 126, 7, 1, 6894720, 3427200, 866880, 141120, 18480, 1680, 168, 8, 1, 123742080, 62052480, 15422400, 2600640

Offset: 0

Paul Barry, Jul 20 2005

			Rows begin
  1;
  1,1;
  6,2,1;
  30,18,3,1;
  264,120,36,4,1;
  2520,1320,300,60,5,1;
  30960,15120,3960,600,90,6,1;

Row sums are A110322.
Antidiagonal sums are A110323.
Inverse is A110324.

T(n,k) = n!*J(n-k+1)/k!, where J(n)=A001045(n).

Column k has e.g.f. (x^k/k!)/(1-x-2x^2).

A110325 Row sums of number triangle related to the Jacobsthal numbers.

Original entry on oeis.org

1, 0, -5, -14, -27, -44, -65, -90, -119, -152, -189, -230, -275, -324, -377, -434, -495, -560, -629, -702, -779, -860, -945, -1034, -1127, -1224, -1325, -1430, -1539, -1652, -1769, -1890, -2015, -2144, -2277, -2414, -2555, -2700, -2849, -3002, -3159, -3320, -3485, -3654, -3827, -4004, -4185, -4370

Offset: 0

Paul Barry, Jul 20 2005

Essentially the same sequence as A014106.

Rows sums of A110324. Results from a general construction: the row sums of the inverse of the number triangle whose columns have e.g.f. (x^k/k!)/(1 - a*x - b*x^2), g.f. (1 - (a+2)*x - (2*b-a-1)*x^2)/(1-x)^3, and general term 1 + (b-a)*n - b*n^2. This is the binomial transform of (1, -a, -2b, 0, 0, 0, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014106 (essentially the same sequence), A110324.

Cf. A005408, A110325.

[1+n-2*n^2: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CoefficientList[Series[(1-3x-2x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
LinearRecurrence[{3,-3,1},{1,0,-5},50] (* Harvey P. Dale, Oct 20 2024 *)

a(n)=1+n-2*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 1 + n - 2*n^2.
G.f.: (1 - 3*x - 2*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
From Elmo R. Oliveira, Nov 02 2024: (Start)
E.g.f.: exp(x)*(1 - x - 2*x^2).
a(n) = -A005408(n)*A110325(n). (End)

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A110326 Diagonal sums of triangle A110324.

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A110321 A Jacobsthal number related number triangle.

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A110325 Row sums of number triangle related to the Jacobsthal numbers.

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