A119329 -id:A119329 - cp's OEIS frontend

A119330 Expansion of (1-x)^2/((1-x)^4-2x^4).

1, 2, 3, 4, 7, 18, 49, 120, 265, 554, 1155, 2476, 5455, 12138, 26881, 58992, 128689, 280466, 612579, 1341268, 2940151, 6443778, 14111857, 30886632, 67590649, 147934010, 323850531, 709047292, 1552412671, 3398703066, 7440375937

Offset: 0

Paul Barry, May 14 2006

Row sums of A119329. Binomial transform of A119332, or (1,1,0,0,2,2,0,0,4,4,...).

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,1).

CoefficientList[Series[(1-x)^2/((1-x)^4-2x^4),{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,1},{1,2,3,4},50] (* Harvey P. Dale, Jul 20 2021 *)

a(n)=4a(n-1)-6a(n-2)+4a(n-3)+a(n-4); a(n)=sum{k=0..n, sum{j=0..n-k, C(k,2j)C(n-k,2j)*2^j}}.

A119331 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)C(n,i)sum{j=0..i-k, C(k,2j)C(i-k,2j)2^j}}.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, -1, 5, -3, 1, 0, 1, -8, 12, -4, 1, 0, -1, 11, -28, 22, -5, 1, 0, 1, -14, 51, -68, 35, -6, 1, 0, -1, 17, -81, 159, -135, 51, -7, 1, 0, 1, -20, 118, -312, 390, -236, 70, -8, 1, 0, -1, 23, -162, 544, -926, 816, -378, 92, -9, 1

Offset: 0

Paul Barry, May 14 2006

Row sums are A119332. Multiply by Pascal's triangle A007318 to get A119329.

			Triangle begins
1,
0, 1,
0, -1, 1,
0, 1, -2, 1,
0, -1, 5, -3, 1,
0, 1, -8, 12, -4, 1,
0, -1, 11, -28, 22, -5, 1,
0, 1, -14, 51, -68, 35, -6, 1,
0, -1, 17, -81, 159, -135, 51, -7, 1,
0, 1, -20, 118, -312, 390, -236, 70, -8, 1,
0, -1, 23, -162, 544, -926, 816, -378, 92, -9, 1

Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,2j)*2^j*x^(2j)}

cp's OEIS Frontend

A119330 Expansion of (1-x)^2/((1-x)^4-2x^4).

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A119331 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)C(n,i)sum{j=0..i-k, C(k,2j)C(i-k,2j)2^j}}.

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cp's OEIS Frontend

A119330 Expansion of (1-x)^2/((1-x)^4-2x^4).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A119331 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i-k, C(k,2j)*C(i-k,2j)*2^j}}.

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Comments

Examples

Formula

A119331 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)C(n,i)sum{j=0..i-k, C(k,2j)C(i-k,2j)2^j}}.