A099511 -id:A099511 - cp's OEIS frontend

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.

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

Paul D. Hanna, Oct 20 2004

Row sums form A099511. 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],
[1,2],
[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.

Cf. A034870, A099509, A099511, A099512.

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

G.f.: (1-x+2*x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2-2*x^3*y^2+x^4*y^4). T(n, k) = binomial(2*n-2*(k\2), k).

A376726 Expansion of (1 + x^2 - x^3)/((1 + x^2 - x^3)^2 - 4*x^2).

Original entry on oeis.org

1, 0, 3, 1, 5, 10, 8, 35, 30, 85, 137, 201, 476, 616, 1357, 2172, 3735, 7193, 11213, 21782, 36064, 64095, 115130, 193769, 354737, 604049, 1074008, 1889968, 3273785, 5839608, 10106859, 17880785, 31325077, 54793282, 96710296, 168730043, 297336790, 520856765, 913684857

Offset: 0

Seiichi Manyama, Oct 03 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,2,-1).

Cf. A099511, A376727, A376728.

Cf. A376723, A376729.

my(N=40, x='x+O('x^N)); Vec((1+x^2-x^3)/((1+x^2-x^3)^2-4*x^2))

a(n) = sum(k=0, n\2, binomial(2*k+1, 2*n-4*k+1));

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

a(n) = Sum_{k=0..floor(n/2)} binomial(2*k+1,2*n-4*k+1).

A376727 Expansion of (1 + x^3 - x^4)/((1 + x^3 - x^4)^2 - 4*x^3).

Original entry on oeis.org

1, 0, 0, 3, 1, 0, 5, 10, 1, 7, 35, 21, 10, 84, 126, 47, 166, 462, 343, 341, 1288, 1731, 1170, 3081, 6453, 5685, 7553, 19572, 25280, 24004, 52789, 93844, 95932, 143435, 299577, 386536, 448673, 873754, 1411193, 1625003, 2536215, 4639077, 6097214, 7959492, 14238226

Offset: 0

Seiichi Manyama, Oct 03 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,2,2,0,-1,2,-1).

Cf. A099511, A376726, A376728.

Cf. A376724, A376730.

my(N=50, x='x+O('x^N)); Vec((1+x^3-x^4)/((1+x^3-x^4)^2-4*x^3))

a(n) = sum(k=0, n\3, binomial(2*k+1, 2*n-6*k+1));

a(n) = 2*a(n-3) + 2*a(n-4) - a(n-6) + 2*a(n-7) - a(n-8).

a(n) = Sum_{k=0..floor(n/3)} binomial(2*k+1,2*n-6*k+1).

A376728 Expansion of (1 + x^4 - x^5)/((1 + x^4 - x^5)^2 - 4*x^4).

Original entry on oeis.org

1, 0, 0, 0, 3, 1, 0, 0, 5, 10, 1, 0, 7, 35, 21, 1, 9, 84, 126, 36, 12, 165, 462, 330, 68, 287, 1287, 1716, 730, 533, 3004, 6435, 5022, 2045, 6293, 19449, 24329, 13345, 14008, 50524, 92400, 76912, 47481, 120156, 294124, 354488, 237139, 299421, 823200, 1354588

Offset: 0

Seiichi Manyama, Oct 03 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,2,0,0,-1,2,-1).

Cf. A099511, A376726, A376727.

Cf. A376725, A376731.

my(N=50, x='x+O('x^N)); Vec((1+x^4-x^5)/((1+x^4-x^5)^2-4*x^4))

a(n) = sum(k=0, n\4, binomial(2*k+1, 2*n-8*k+1));

a(n) = 2*a(n-4) + 2*a(n-5) - a(n-8) + 2*a(n-9) - a(n-10).

a(n) = Sum_{k=0..floor(n/4)} binomial(2*k+1,2*n-8*k+1).

A376785 Expansion of (1 + x - x^3)/((1 + x - x^3)^2 - 4*x).

Original entry on oeis.org

1, 3, 5, 8, 19, 46, 98, 201, 429, 937, 2024, 4325, 9260, 19916, 42841, 91999, 197485, 424160, 911255, 1957402, 4203998, 9029425, 19394681, 41658577, 89478064, 192188361, 412801176, 886657848, 1904452689, 4090568027, 8786123349, 18871711384, 40534539675, 87064092870

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,2,0,-1).

Cf. A099511, A376786.

Cf. A375278, A375279, A376717.

CoefficientList[Series[(1+x-x^3)/((1+x-x^3)^2-4x),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2,2,0,-1},{1,3,5,8,19,46},40] (* Harvey P. Dale, Jun 29 2025 *)

my(N=40, x='x+O('x^N)); Vec((1+x-x^3)/((1+x-x^3)^2-4*x))

a(n) = sum(k=0, n\3, binomial(2*n-4*k+1, 2*k+1));

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k+1,2*k+1).

A376786 Expansion of (1 + x - x^4)/((1 + x - x^4)^2 - 4*x).

Original entry on oeis.org

1, 3, 5, 7, 10, 21, 48, 99, 183, 326, 602, 1165, 2282, 4396, 8318, 15675, 29743, 56841, 108765, 207510, 394809, 750880, 1429845, 2725685, 5196420, 9901692, 18859649, 35921156, 68432064, 130388316, 248437405, 473322419, 901717453, 1717851555, 3272777450

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,2,0,0,-1).

Cf. A099511, A376785.

Cf. A375282, A375283, A376718.

my(N=40, x='x+O('x^N)); Vec((1+x-x^4)/((1+x-x^4)^2-4*x))

a(n) = sum(k=0, n\4, binomial(2*n-6*k+1, 2*k+1));

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) + 2*a(n-5) - a(n-8).

a(n) = Sum_{k=0..floor(n/4)} binomial(2*n-6*k+1,2*k+1).

A387627 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(2n-2k+1,2*k+1).

Original entry on oeis.org

1, 3, 7, 27, 83, 263, 855, 2723, 8731, 27999, 89663, 287355, 920771, 2950263, 9453607, 30291667, 97062123, 311012623, 996563855, 3193247403, 10231988371, 32785923879, 105054547063, 336621829635, 1078623042491, 3456186066623, 11074510391007, 35485583833307

Offset: 0

Seiichi Manyama, Sep 03 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,4,-4).

Cf. A001653, A387628, A387629.

Cf. A099511.

[&+[2^k* Binomial(2*n-2*k+1, 2*k+1): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Table[Sum[2^k*Binomial[2*n-2*k+1,2*k+1],{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 04 2025 *)

a(n) = sum(k=0, n\2, 2^k*binomial(2*n-2*k+1, 2*k+1));

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

a(n) = 2*a(n-1) + 3*a(n-2) + 4*a(n-3) - 4*a(n-4).

cp's OEIS Frontend

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.

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A376726 Expansion of (1 + x^2 - x^3)/((1 + x^2 - x^3)^2 - 4*x^2).

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A376727 Expansion of (1 + x^3 - x^4)/((1 + x^3 - x^4)^2 - 4*x^3).

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A376728 Expansion of (1 + x^4 - x^5)/((1 + x^4 - x^5)^2 - 4*x^4).

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A376785 Expansion of (1 + x - x^3)/((1 + x - x^3)^2 - 4*x).

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A376786 Expansion of (1 + x - x^4)/((1 + x - x^4)^2 - 4*x).

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A387627 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(2*n-2*k+1,2*k+1).

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A387627 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(2n-2k+1,2*k+1).