A099602 -id:A099602 - cp's OEIS frontend

A099603 Row sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

1, 2, 4, 12, 20, 64, 104, 336, 544, 1760, 2848, 9216, 14912, 48256, 78080, 252672, 408832, 1323008, 2140672, 6927360, 11208704, 36272128, 58689536, 189923328, 307302400, 994451456, 1609056256, 5207015424, 8425127936, 27264286720, 44114542592, 142757658624, 230986743808

Offset: 0

Paul D. Hanna, Oct 25 2004

			Sequence begins: {1*1, 1*2, 2*2, 3*4, 5*4, 8*8, 13*8, 21*16, 34*16, ...}.

Stefano Spezia, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-4).

Cf. A000045, A027907, A099602.

LinearRecurrence[{0,6,0,-4},{1,2,4,12},30] (* Harvey P. Dale, Aug 09 2016 *)

PARI
```
a(n)=fibonacci(n+1)*2^((n+1)\2)
```

a(n) = Fibonacci(n+1)*2^((n+1)/2).
a(n) = 6*a(n-2) - 4*a(n-4) for n>4.
G.f.: (1+2*x-2*x^2)/(1-6*x^2+4*x^4).

A099604 Antidiagonal sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 23, 40, 72, 131, 233, 420, 756, 1355, 2438, 4381, 7868, 14144, 25413, 45661, 82058, 147444, 264943, 476092, 855483, 1537236, 2762296, 4963591, 8919173, 16027012, 28799164, 51749715, 92989886, 167094985, 300255720

Offset: 0

Paul D. Hanna, Oct 25 2004

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 2, 3, 0, -2, -1).

Cf. A099602, A027907.

LinearRecurrence[{0, 2, 3, 0, -2, -1}, {1, 1, 2, 4, 7, 12}, 35] (* Jean-François Alcover, Oct 30 2017 *)

a(n)=polcoeff((1+x-x^3)/(1-2*x^2-3*x^3+2*x^5+x^6)+x*O(x^n),n,x)

G.f.: (1+x-x^3)/(1-2*x^2-3*x^3+2*x^5+x^6).

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

A104495 Matrix inverse of triangle A099602, read by rows, where row n of A099602 equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 3, -4, 1, 1, -4, 12, -5, 1, -1, 5, -34, 17, -7, 1, 1, -6, 98, -51, 32, -8, 1, -1, 7, -294, 149, -124, 40, -10, 1, 1, -8, 919, -443, 448, -164, 61, -11, 1, -1, 9, -2974, 1362, -1576, 612, -298, 72, -13, 1, 1, -10, 9891, -4336, 5510, -2188, 1294, -370, 99, -14, 1, -1, 11, -33604, 14227, -19322, 7698

Offset: 0

Paul D. Hanna, Mar 11 2005

Row sums are A104496. Absolute row sums form A014137 (partial sums of Catalan numbers). Column 2 is signed A014143.

			Rows begin:
1;
-1,1;
1,-2,1;
-1,3,-4,1;
1,-4,12,-5,1;
-1,5,-34,17,-7,1;
1,-6,98,-51,32,-8,1;
-1,7,-294,149,-124,40,-10,1;
1,-8,919,-443,448,-164,61,-11,1;
-1,9,-2974,1362,-1576,612,-298,72,-13,1; ...

Cf. A099602, A027907, A000108, A104496, A014137, A014143.

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( (1+X*Y/(1+X))/(1+X-Y^2*(1-(1+4*X)^(1/2))^2/4),n,x),k,y)}

G.f.: A(x, y) = (1 + x*y/(1+x))/(1+x - x^2*y^2*Catalan(-x)^2), also G.f.: Column_k(x) = Catalan(-x)^(2*[k/2])/(1+x)^[(k+3)/2], where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).

A099605 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle A034870 of coefficients in successive powers of the trinomial (1+2*x+x^2), omitting leading zeros.

Original entry on oeis.org

1, 2, 2, 1, 5, 4, 4, 16, 20, 8, 1, 14, 41, 44, 16, 6, 50, 146, 198, 128, 32, 1, 27, 155, 377, 456, 272, 64, 8, 112, 560, 1408, 1992, 1616, 704, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 10, 210, 1572, 6084, 14002, 20330, 18880, 10912, 3584, 512, 1, 65

Offset: 0

Paul D. Hanna, Oct 25 2004

Row sums form A099606, where A099606(n) = Pell(n+1)*2^[(n+1)/2]. Central coefficients of even-indexed rows form A026000, where A026000(n) = T(2n,n), where T = Delannoy triangle (A008288). Antidiagonal sums form A099607.

			Rows begin:
[1],
[2,2],
[1,5,4],
[4,16,20,8],
[1,14,41,44,16],
[6,50,146,198,128,32],
[1,27,155,377,456,272,64],
[8,112,560,1408,1992,1616,704,128],
[1,44,406,1652,3649,4712,3568,1472,256],
[10,210,1572,6084,14002,20330,18880,10912,3584,512],
[1,65,870,5202,17469,36365,48940,42800,23552,7424,1024],...
The binomial transform of row 2 equals column 2 of A034870:
BINOMIAL[1,5,4] = [1,6,15,28,45,66,91,120,153,...].
The binomial transform of row 3 equals column 3 of A034870:
BINOMIAL[4,16,20,8] = [4,20,56,120,220,364,560,...].
The binomial transform of row 4 equals column 4 of A034870:
BINOMIAL[1,14,41,44,16] = [1,15,70,210,495,1001,...].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A099602, A099605, A034870, A000129.

CoefficientList[CoefficientList[Series[(1 + 2*(y + 1)*x - (y + 1)*x^2)/(1 - (2*y + 1)*(2*y + 2)*x^2 + (y + 1)^2*x^4), {x, 0, 49}, {y, 0, 49}], x],
  y] // Flatten (* G. C. Greubel, Apr 14 2017 *)

{T(n,k)=polcoeff(polcoeff((1+2*(y+1)*x-(y+1)*x^2)/(1-(2*y+1)*(2*y+2)*x^2+(y+1)^2*x^4)+x*O(x^n),n,x)+y*O(y^k),k,y)}

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

A104496 Expansion of 2(2x+1)/((x+1)(sqrt(4x+1)+1)).

Original entry on oeis.org

1, 0, 0, -1, 5, -19, 67, -232, 804, -2806, 9878, -35072, 125512, -452388, 1641028, -5986993, 21954973, -80884423, 299233543, -1111219333, 4140813373, -15478839553, 58028869153, -218123355523, 821908275547, -3104046382351, 11747506651599, -44546351423299, 169227201341651

Offset: 0

Paul D. Hanna, Mar 11 2005

Previous name was: Row sums of triangle A104495. A104495 equals the matrix inverse of triangle A099602, where row n of A099602 equals the inverse Binomial transform of column n of the triangle of trinomial coefficients (A027907).

Absolute row sums of triangle A104495 forms A014137 (partial sums of Catalan numbers).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A104495, A099602, A027907, A000108.

gf := (2*(2*x+1))/((x+1)*(sqrt(4*x+1)+1)): ser := series(gf,x,30):
seq(coeff(ser,x,n),n=0..28); # Peter Luschny, Apr 25 2016

CoefficientList[Series[(1+2*x)/(1+x)/(1+x - (1-(1+4*x)^(1/2))^2/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=local(X=x+x*O(x^n));polcoeff( (1+2*X)/(1+X)/(1+X-(1-(1+4*X)^(1/2))^2/4),n,x)}

from itertools import accumulate
def A104496_list(size):
    if size < 1: return []
    L, accu = [1], [1]
    for n in range(size-1):
        accu = list(accumulate(accu + [-accu[0]]))
        L.append(-(-1)**n*accu[-1])
    return L
print(A104496_list(29)) # Peter Luschny, Apr 25 2016

G.f.: A(x) = (1 + 2*x)/(1+x)/(1+x - x^2*Catalan(-x)^2), where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).
a(n) ~ (-1)^n * 2^(2*n+1) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014
D-finite with recurrence: (n+1)*a(n) +(7*n-3)*a(n-1) +2*(7*n-12)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jan 23 2020

New name using the g.f. of the author by Peter Luschny, Apr 25 2016

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A099603 Row sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

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A099604 Antidiagonal sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

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A104495 Matrix inverse of triangle A099602, read by rows, where row n of A099602 equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

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A099605 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle A034870 of coefficients in successive powers of the trinomial (1+2*x+x^2), omitting leading zeros.

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A104496 Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).

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A104496 Expansion of 2(2x+1)/((x+1)(sqrt(4x+1)+1)).