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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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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
FORMULA
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).
EXAMPLE
Sequence begins: {1*1, 1*2, 2*2, 3*4, 5*4, 8*8, 13*8, 21*16, 34*16, ...}.
MATHEMATICA
LinearRecurrence[{0, 6, 0, -4}, {1, 2, 4, 12}, 30] (* Harvey P. Dale, Aug 09 2016 *)
PROG
(PARI) a(n)=fibonacci(n+1)*2^((n+1)\2)
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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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
FORMULA
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.
MATHEMATICA
LinearRecurrence[{0, 2, 3, 0, -2, -1}, {1, 1, 2, 4, 7, 12}, 35] (* Jean-François Alcover, Oct 30 2017 *)
PROG
(PARI) 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)
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).
+20
2
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
COMMENTS
Row sums are A104496. Absolute row sums form A014137 (partial sums of Catalan numbers). Column 2 is signed A014143.
FORMULA
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).
EXAMPLE
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; ...
PROG
(PARI) {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)}
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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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
COMMENTS
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.
FORMULA
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.
EXAMPLE
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,...].
MATHEMATICA
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],
PROG
(PARI) {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)}
Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).
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3
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
COMMENTS
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).
FORMULA
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
MAPLE
gf := (2*(2*x+1))/((x+1)*(sqrt(4*x+1)+1)): ser := series(gf, x, 30):
MATHEMATICA
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 *)
PROG
(PARI) {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)}
(Python)
from itertools import accumulate
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
EXTENSIONS
New name using the g.f. of the author by Peter Luschny, Apr 25 2016
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