A099510 -id:A099510 - cp's OEIS frontend

A099511 Row sums of triangle A099510, so that a(n) = Sum_{k=0..n} coefficient of z^k in (1 + 2*z + z^2)^(n-[k/2]), where [k/2] is the integer floor of k/2.

1, 3, 6, 17, 45, 116, 305, 799, 2090, 5473, 14329, 37512, 98209, 257115, 673134, 1762289, 4613733, 12078908, 31622993, 82790071, 216747218, 567451585, 1485607537, 3889371024, 10182505537, 26658145587, 69791931222, 182717648081

Offset: 0

Paul D. Hanna, Oct 21 2004

nonn

Cf. A099510.

a(n)=sum(k=0,n,polcoeff((1+2*x+x^2+x*O(x^k))^(n-k\2),k))

G.f.: (1+x-x^2)/(1-2*x-x^2-2*x^3+x^4). a(n) = Sum_{k=0..n} binomial(2*n-2*[k/2], k).

A099509 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 + 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, 1, 1, 3, 3, 2, 1, 4, 6, 7, 1, 1, 5, 10, 16, 6, 3, 1, 6, 15, 30, 19, 16, 1, 1, 7, 21, 50, 45, 51, 10, 4, 1, 8, 28, 77, 90, 126, 45, 30, 1, 1, 9, 36, 112, 161, 266, 141, 126, 15, 5, 1, 10, 45, 156, 266, 504, 357, 393, 90, 50, 1, 1, 11, 55, 210, 414, 882, 784, 1016, 357

Offset: 0

Paul D. Hanna, Oct 20 2004

Row sums form absolute values of A078039. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).

			Rows begin:
[1],
[1,1],
[1,2,1],
[1,3,3,2],
[1,4,6,7,1],
[1,5,10,16,6,3],
[1,6,15,30,19,16,1],
[1,7,21,50,45,51,10,4],
[1,8,28,77,90,126,45,30,1],
[1,9,36,112,161,266,141,126,15,5],...
and can be derived from coefficients of (1+z+z^2)^n:
[1],
[1,1,1],
[1,2,3,2,1],
[1,3,6,7,6,3,1],
[1,4,10,16,19,16,10,4,1],
[1,5,15,30,45,51,45,30,15,5,1],...
by shifting each column k down by [k/2] rows.

Cf. A027907, A078039, A099510.

PARI
```
T(n,k)=if(n
				
```

G.f.: (1-x+x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2+x^3*y^2+x^4*y^4).

A099512 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 + 3*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, 3, 1, 6, 1, 1, 9, 11, 6, 1, 12, 30, 45, 1, 1, 15, 58, 144, 30, 9, 1, 18, 95, 330, 195, 144, 1, 1, 21, 141, 630, 685, 873, 58, 12, 1, 24, 196, 1071, 1770, 3258, 685, 330, 1, 1, 27, 260, 1680, 3801, 9198, 3989, 3258, 95, 15, 1, 30, 333, 2484, 7210, 21672, 15533

Offset: 0

Paul D. Hanna, Oct 20 2004

Row sums form A099513. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).

			Rows begin:
[1],
[1,3],
[1,6,1],
[1,9,11,6],
[1,12,30,45,1],
[1,15,58,144,30,9],
[1,18,95,330,195,144,1],
[1,21,141,630,685,873,58,12],
[1,24,196,1071,1770,3258,685,330,1],
[1,27,260,1680,3801,9198,3989,3258,95,15],...
and can be derived from coefficients of (1+3*z+z^2)^n:
[1],
[1,3,1],
[1,6,11,6,1],
[1,9,30,45,30,9,1],
[1,12,58,144,195,144,58,12,1],
[1,15,95,330,685,873,685,330,95,15,1],...
by shifting each column k down by [k/2] rows.

Cf. A099509, A099510, A099513.

PARI
```
T(n,k)=if(n
				
```

G.f.: (1-x+3*x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2-7*x^3*y^2+x^4*y^4).

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

Paul D. Hanna, Oct 20 2004

Row sums form A099515. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).

			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.

Cf. A099509, A099510, A099512, A099515.

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 *)

PARI
```
T(n,k)=if(n
				
```

G.f.: (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).

a(50..51) corrected by Georg Fischer, Feb 17 2020

A099527 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 (2 + 3*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, 2, 3, 4, 12, 1, 8, 36, 13, 6, 16, 96, 66, 63, 1, 32, 240, 248, 360, 33, 9, 64, 576, 800, 1560, 321, 180, 1, 128, 1344, 2352, 5760, 1970, 1683, 62, 12, 256, 3072, 6496, 19152, 9420, 10836, 985, 390, 1, 512, 6912, 17152, 59136, 38472, 55692, 8989, 5418, 100, 15

Offset: 0

Paul D. Hanna, Oct 20 2004

Row sums form A099528. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).

			Rows begin:
[1],
[2,3],
[4,12,1],
[8,36,13,6],
[16,96,66,63,1],
[32,240,248,360,33,9],
[64,576,800,1560,321,180,1],
[128,1344,2352,5760,1970,1683,62,12],
[256,3072,6496,19152,9420,10836,985,390,1],
[512,6912,17152,59136,38472,55692,8989,5418,100,15],...
and can be derived from the coefficients of (2+3*z+z^2)^n:
[1],
[2,3,1],
[4,12,13,6,1],
[8,36,66,63,33,9,1],
[16,96,248,360,321,180,62,12,1],
[32,240,800,1560,1970,1683,985,390,100,15,1],...
by shifting each column k down by [k/2] rows.

Cf. A099509, A099510, A099528.

PARI
```
T(n,k)=if(n
				
```

G.f.: (1-x*(2-3*y)-x^2*y^2)/(1-4*x+x^2*(4-2*y^2)-5*x^3*y^2+x^4*y^4).

cp's OEIS Frontend

A099511 Row sums of triangle A099510, so that a(n) = Sum_{k=0..n} coefficient of z^k in (1 + 2*z + z^2)^(n-[k/2]), where [k/2] is the integer floor of k/2.

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A099509 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 + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.

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A099512 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 + 3*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.

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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.

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Mathematica

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A099527 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 (2 + 3*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.

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Author

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Comments

Examples

Crossrefs

Programs

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