A099513
Row sums of triangle A099512, so that a(n) = Sum_{k=0..n} coefficient of z^k in (1 + 3*z + z^2)^(n-[k/2]), where [k/2] is the integer floor of k/2.
Original entry on oeis.org
1, 4, 8, 27, 89, 257, 784, 2421, 7336, 22324, 68147, 207549, 632177, 1926608, 5870089, 17884476, 54493120, 166034731, 505883825, 1541369745, 4696373312, 14309268413, 43598614528, 132839740908, 404746601923, 1233213978037
Offset: 0
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LinearRecurrence[{2,1,7,-1},{1,4,8,27},30] (* or *) CoefficientList[ Series[ (1+2x-x^2)/(1-2x-x^2-7x^3+x^4),{x,0,30}],x] (* Harvey P. Dale, Jul 12 2011 *)
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a(n)=sum(k=0,n,polcoeff((1+3*x+x^2+x*O(x^k))^(n-k\2),k))
A099510
Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + 2*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.
Original entry on oeis.org
1, 1, 2, 1, 4, 1, 1, 6, 6, 4, 1, 8, 15, 20, 1, 1, 10, 28, 56, 15, 6, 1, 12, 45, 120, 70, 56, 1, 1, 14, 66, 220, 210, 252, 28, 8, 1, 16, 91, 364, 495, 792, 210, 120, 1, 1, 18, 120, 560, 1001, 2002, 924, 792, 45, 10, 1, 20, 153, 816, 1820, 4368, 3003, 3432, 495, 220, 1, 1, 22, 190
Offset: 0
Rows begin:
[1],
[1,2],
[1,4,1],
[1,6,6,4],
[1,8,15,20,1],
[1,10,28,56,15,6],
[1,12,45,120,70,56,1],
[1,14,66,220,210,252,28,8],
[1,16,91,364,495,792,210,120,1],
[1,18,120,560,1001,2002,924,792,45,10],...
and can be derived from coefficients of (1+2*z+z^2)^n:
[1],
[1,2,1],
[1,4,6,4,1],
[1,6,15,20,15,6,1],
[1,8,28,56,70,56,28,8,1],
[1,10,45,120,210,252,210,120,45,10,1],...
by shifting each column k down by [k/2] rows.
A099514
Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + z + 2*z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 13, 4, 1, 5, 14, 28, 18, 12, 1, 6, 20, 50, 49, 56, 8, 1, 7, 27, 80, 105, 161, 56, 32, 1, 8, 35, 119, 195, 366, 210, 200, 16, 1, 9, 44, 168, 329, 721, 581, 732, 160, 80, 1, 10, 54, 228, 518, 1288, 1337, 2045, 780, 640, 32, 1, 11, 65, 300, 774, 2142, 2716, 4824, 2674, 2884, 432, 192
Offset: 0
Rows begin:
[1],
[1,1],
[1,2,2],
[1,3,5,4],
[1,4,9,13,4],
[1,5,14,28,18,12],
[1,6,20,50,49,56,8],
[1,7,27,80,105,161,56,32],
[1,8,35,119,195,366,210,200,16],
[1,9,44,168,329,721,581,732,160,80],...
and can be derived from coefficients of (1+z+2*z^2)^n:
[1],
[1,1,2],
[1,2,5,4,4],
[1,3,9,13,18,12,8],
[1,4,14,28,49,56,56,32,16],
[1,5,20,50,105,161,210,200,160,80,32],...
by shifting each column k down by [k/2] rows.
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With[{m = 11}, CoefficientList[CoefficientList[Series[(1-x+x*y-2*x^2*y^2)/ ((1-x)^2-4*x^2*y^2+3*x^3*y^2+4*x^4*y^4), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)
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T(n,k)=if(n
Showing 1-3 of 3 results.
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