A055249 -id:A055249 - cp's OEIS frontend

A055583 Seventh column of triangle A055252.

1, 10, 58, 256, 955, 3178, 9740, 28064, 77093, 203930, 523262, 1309520, 3209871, 7731642, 18348240, 42989520, 99612345, 228586890, 520090690, 1174401760, 2634019171, 5872021450, 13019115028, 28722588736, 63082326605

Offset: 0

Wolfdieter Lang, May 26 2000

Michael De Vlieger, Table of n, a(n) for n = 0..3295
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (10,-42,96,-129,102,-44,8).

Cf. A055252, A055249, A055250, partial sums of A055582.

CoefficientList[Series[1/(((1 - 2 x)^3) (1 - x)^4), {x, 0, 24}], x] (* Michael De Vlieger, Apr 24 2020 *)
LinearRecurrence[{10,-42,96,-129,102,-44,8},{1,10,58,256,955,3178,9740},30] (* Harvey P. Dale, Nov 06 2022 *)

G.f.: 1/(((1-2*x)^3)*(1-x)^4).
a(n) = A055252(n+6, 6).
a(n) = Sum_{j=0..n-1} a(j) + A055250(n), n >= 1.

A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 2, 2, 1, 3, 2, 3, 3, 1, 0, 3, 4, 5, 4, 1, 4, 3, 5, 7, 8, 5, 1, 0, 4, 6, 9, 12, 12, 6, 1, 5, 4, 7, 11, 16, 20, 17, 7, 1, 0, 5, 8, 13, 20, 28, 32, 23, 8, 1, 6, 5, 9, 15, 24, 36, 48, 49, 30, 9, 1, 0, 6, 10, 17, 28, 44, 64, 80, 72, 38, 10, 1, 7, 6, 11, 19, 32, 52, 80, 112, 129

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n,0)=T(n,2) by T(2n,0)=T(n,m) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058393, A058395, A057884 (and effectively A007318).

			Rows are (1,0,2,0,3,0,4,...), (1,1,2,2,3,3,...), (1,2,3,4,5,6,...), (1,3,5,7,9,11,...), etc.

Rows are A027656 (A000027 with zeros), A008619, A000027, A005408, A008574 etc. Columns are A000012, A001477, A022856 etc. Diagonals include A034007, A045891, A045623, A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 etc. The triangle A055249 also appears in half of the array.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(2n, 0)=T(n, 2) and T(2n+1, 0)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^2.

A106194 Triangle read by rows, generated from binomial transforms of odd numbers.

Original entry on oeis.org

1, 4, 1, 12, 5, 1, 32, 17, 6, 1, 80, 49, 23, 7, 1, 192, 129, 72, 30, 8, 1, 448, 321, 201, 102, 38, 9, 1, 1024, 769, 522, 303, 140, 47, 10, 1, 2304, 1793, 1291, 825, 443, 187, 57, 11, 1, 5120, 4097, 3084, 2116, 1268, 630, 244, 68, 12, 1

Offset: 0

Gary W. Adamson, Apr 24 2005

Appending the binomial transform of the natural numbers, (A001792: 1, 3, 8, 20, 48...) to A106194 as a leftmost column creates triangle A055249.
Placing zeros into the offset spaces, column 1: 0, 1, 5, 17, 49...; is the binomial transform of 0, 1, 3, 5...; and alternatively the binomial transform of 0, 0, 1, 2, 3...
n-th column is the binomial transform of 1, 3, 5...prefaced by n zeros. n-th column is alternatively the binomial transform of 1, 2, 3...prefaced by (n+1) zeros. The triangle of A106194 is identical to the binomial transform (of natural numbers, prefaced with zeros) triangle: A055249, deleting the leftmost column.

			First few rows of the triangle are:
1;
4, 1;
12, 5, 1;
32, 17, 6, 1;
80, 49, 23, 7, 1;
192, 129, 72, 30, 8, 1;
448, 321, 201, 102, 38, 9, 1;
...

Cf. A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055249.

cp's OEIS Frontend

A055583 Seventh column of triangle A055252.

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A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

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Formula

A106194 Triangle read by rows, generated from binomial transforms of odd numbers.

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Examples

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