A084860 -id:A084860 - cp's OEIS frontend

A084861 Expansion of (1-3x+4x^2-3x^3+x^4)/(1-2x)^2.

1, 1, 4, 9, 21, 48, 108, 240, 528, 1152, 2496, 5376, 11520, 24576, 52224, 110592, 233472, 491520, 1032192, 2162688, 4521984, 9437184, 19660800, 40894464, 84934656, 176160768, 364904448, 754974720, 1560281088, 3221225472

Offset: 0

Paul Barry, Jun 12 2003

Partial sums give A084860. Binomial transform of signed version of A008795.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

CoefficientList[Series[(1-3x+4x^2-3x^3+x^4)/(1-2x)^2,{x,0,40}],x] (* or *) LinearRecurrence[{4,-4},{1,1,4,9,21},40] (* Harvey P. Dale, Nov 25 2020 *)

a(n) = 3(n+3)2^(n-4), n>2.

A139633 Triangle read by rows: binomial transform of a diagonalized matrix of A026741.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 9, 2, 1, 4, 18, 8, 5, 1, 5, 30, 20, 25, 3, 1, 6, 45, 40, 75, 18, 7, 1, 7, 63, 70, 175, 63, 49, 4, 1, 8, 84, 112, 350, 168, 196, 32, 9, 1, 9, 108, 168, 630, 378, 588, 144, 81, 5, 1, 10, 135, 240, 1050, 756, 1470, 480, 405, 50, 11, 1, 11, 165, 330, 1650, 1386

Offset: 1

Gary W. Adamson and Roger L. Bagula, Apr 27 2008

Row sums = A084860: (1, 2, 6, 15, 36, 84, 192,...).

			First few rows of the triangle are:
1;
1, 1;
1, 2, 3;
1, 3, 9, 2;
1, 4, 18, 8, 5;
1, 5, 30, 20, 25, 3;
1, 6, 45, 40, 75, 18, 7;
1, 7, 63, 70, 175, 63, 49, 4;
...

Cf. A026741, A084860.

Let X = a diagonalized matrix of A026741: [1; 0,1; 0,0,3; 0,0,0,2;], where the first few nonzero terms of A026741 are (1, 1, 3, 2, 5, 3, 7,...). The triangle = A007318 * X.

A129566 A007318 * A129565.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 6, 4, 1, 8, 11, 11, 5, 1, 16, 20, 25, 16, 6, 1, 32, 37, 51, 42, 22, 7, 1, 64, 70, 98, 98, 64, 29, 8, 1, 128, 135, 183, 211, 163, 93, 37, 9, 1, 256, 264, 339, 429, 381, 256, 130, 46, 10, 1

Offset: 1

Gary W. Adamson, Apr 21 2007

Row sums = A084860: (1, 2, 6, 15, 36, 84, 192, ...).

			First few rows of the triangle:
   1;
   1,  1;
   2,  3,  1;
   4,  6,  4,  1;
   8, 11, 11,  5,  1;
  16, 20, 25, 16,  6,  1;
  32, 37, 51, 42, 22,  7,  1;
  ...

Cf. A007318, A129565, A084860.

Binomial transform of A129565.

cp's OEIS Frontend

A084861 Expansion of (1-3x+4x^2-3x^3+x^4)/(1-2x)^2.

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A139633 Triangle read by rows: binomial transform of a diagonalized matrix of A026741.

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A129566 A007318 * A129565.

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