A105306 -id:A105306 - cp's OEIS frontend

A195304 Decimal expansion of shortest length of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5).

1, 8, 9, 6, 3, 0, 0, 5, 6, 6, 3, 0, 9, 2, 0, 2, 0, 1, 4, 7, 5, 3, 8, 6, 7, 2, 0, 3, 6, 5, 4, 8, 1, 9, 9, 1, 7, 0, 8, 0, 1, 0, 3, 2, 8, 2, 9, 8, 1, 9, 2, 8, 6, 6, 6, 4, 1, 0, 2, 7, 8, 4, 3, 9, 4, 4, 4, 2, 9, 7, 6, 3, 7, 7, 2, 5, 4, 6, 2, 9, 2, 1, 1, 7, 4, 3, 4, 9, 5, 1, 7, 5, 2, 6, 6, 7, 2, 1, 0, 7

Offset: 1

Clark Kimberling, Sep 18 2011

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P.  Philo lines are not generally Euclidean-constructible.
...
Suppose that P lies inside a triangle ABC.  Let (A) denote the shortest length of segment from AB through P to AC, and likewise for (B) and (C).  The Philo sum for ABC and P is here introduced as s=(A)+(B)+(C), and the Philo number for ABC and P, as s/(a+b+c), denoted by Philo(ABC,P).
...
Listed below are examples for which P=G (the centroid); in this list, r'n means sqrt(n) and t=(1+sqrt(5))/2 (the golden ratio).
a....b...c........(A).......(B)........(C)...Philo(ABC,G)
3....4....5......A195304...A195305....A105306...A195411
5....12...13.....A195412...A195413....A195414...A195424
7....24...25.....A195425...A195426....A195427...A195428
8....15...17.....A195429...A195430....A195431...A195432
1....1....r'2....A195433..-1+A179587..A195433...A195436
1....2....r'5....A195434...A195435....A195444...A195445
1....3....r'10...A195446...A195447....A195448...A195449
2....3....r'13...A195450...A195451....A195452...A195453
r'2..r'3..r'5....A195454...A195455....A195456...A195457
1....r'2..r'3....A195471...A195472....A195473...A195474
1....r'3..2......A195475...A195476....A195477...A195478
2....r'5..3......A195479...A195480....A195481...A195482
r'2..r'5..r'7....A195483...A195484....A195485...A195486
r'7..3....4......A195487...A195488....A195489...A195490
1....r't..t......A195491...A195492....A195493...A195494
t-1..t....r'3....A195495...A195496....A195497...A195498
A similar list for P=incenter is given at A195284.

			1.89630056630920201475386720365481991708010328....

Index entries for algebraic numbers, degree 6

Cf. A195305, A195306, A195307; A195284 (P=incenter).

a = 3; b = 4; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100]   (* (A) A195304 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100]   (* (B) A195305 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100]   (* (C) A195306 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)

polrootsreal(2025*x^6 + 21429*x^4 + 4939*x^2 - 389017)[2] \\ Charles R Greathouse IV, Feb 03 2025

A077998 Expansion of (1-x)/(1-2*x-x^2+x^3).

Original entry on oeis.org

1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209, 19181424995, 43100270734, 96845429254

Offset: 0

N. J. A. Sloane, Nov 17 2002

Let u(k), v(k), w(k) be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (A006356 with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (A006054 with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = this sequence with an extra initial 0. - Benoit Cloitre, Apr 05 2002 [Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary W. Adamson, Dec 23 2003]
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A077998 counts closed walks of length n at the vertex of degree 4. - Paul Barry, Oct 02 2004
a(n) is the number of Motzkin (n+2)-sequences with no flatsteps at ground level and whose height is <=2. For example, a(3)=6 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD, UUFDD. - David Callan, Dec 09 2004
Number of compositions of n if there are two kinds of part 2. Example: a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1). Row sums of A105477. - Emeric Deutsch, Apr 09 2005
Diagonal sums of A056242. - Paul Barry, Dec 26 2007
Diagonal sums of triangle in A105306. - Philippe Deléham, Nov 16 2008
a(n) appears in the formula for the nonpositive powers of rho:= 2*cos(Pi/7), the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with sigma:=rho^2-1, the ratio of the larger heptagon diagonal to the side length, as follows. rho^(-n) = a(n)*1 + a(n-1)*rho - C(n)*sigma, n>=0, with C(n)=A006054(n+1). Put a(-1):=0. See the Steinbach reference, and a comment under A052547.
The limit a(n+1)/a(n) for n -> infinity is sigma = rho^2-1, approximately 2.246979603. See a Nov 07 2013 comment on A006054 for the proof, and the preceding comment for rho and sigma and the P. Steinbach reference. - Wolfdieter Lang, Nov 07 2013
From Greg Dresden and Aaron Zhou, Jun 15 2023: (Start)
a(n) is the number of ways to tile a skew double-strip of 3*n cells using all possible "trominos". Here is the skew double-strip corresponding to n=4, with 12 cells:
   _ ___ _ ___ _ ___
  |   |   |   |   |   |   |
 |__|___|_|___| |___|
|   |   |   |   |   |   |
|_|___|_|___|_|___|,
and here are the three possible "tromino" tiles, which can be rotated or reflected as needed:
   _              _
  |   |            |   |
 |__|_      ___|___|     _________
|   |   |    |   |   |      |   |   |   |
|_|___|,   |_|___|  ,   |_|___|_|.
As an example, here is one of the a(4) = 14 ways to tile the skew double-strip of 12 cells:
   _ ___ _____ _______
  |   |   |       |       |
 |   |  |___  |     |
|       |       |   |   |
|_____|_______|_|___|. (End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 14*x^4 + 31*x^5 + 70*x^6 + 157*x^7 + 353*x^8 + ... - _Michael Somos_, Dec 12 2023

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z. 281 (2015) 1061.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015.
Floor van Lamoen, Wave sequences
R. Witula, D. Slota, and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Apart from initial term, same as A006356, which is the main entry for this sequence. A106803 is yet another version.

Cf. A096976, A105477.

a:=[1,1,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jun 27 2019

I:=[1,1,3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 01 2017

CoefficientList[Series[(1-x)/(1-2*x-x^2+x^3), {x, 0, 40}], x] (* Stefan Steinerberger, Sep 11 2006 *)
LinearRecurrence[{2,1,-1},{1,1,3},40] (* Roman Witula, Aug 07 2012 *)
a[ n_] := {1, 0, 0} . MatrixPower[{{0, 1, 0}, {0, 0, 1}, {-1, 1, 2}}, n] . {1, 1, 3}; (* Michael Somos, Dec 12 2023 *)

a(n)=([0,1,0; 0,0,1; -1,1,2]^n*[1;1;3])[1,1] \\ Charles R Greathouse IV, May 10 2016

((1-x)/(1-2*x-x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

a(0)=a(1)=1, a(2)=3, a(n+1) = 2*a(n) + a(n-1) - a(n-2) for n>=2. - Philippe Deléham, Sep 07 2006
7*a(n) = (s(2))^2*(1+c(1))^n + (s(4))^2*(1+c(2))^n + (s(1))^2(1+c(4))^n, where c(j) = 2*Cos(2Pi*j/7) and s(j) = 2*Sin(2Pi*j/7) - for the proof of this one and many other relations for the sequences u(k), v(k) and w(k) defined on the top of the comments by Benoit Cloitre - see Witula et al.'s paper. - Roman Witula, Aug 07 2012
a(n) = b(n+2)- b(n+1), first differences of b(n) = A006054(n). - Wolfdieter Lang, Nov 07 2013; corrected by Kai Wang, May 31 2017
a(n) = A096976(-n) for all n in Z. - Michael Somos, Dec 12 2023

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar

A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 8, 1, 5, 14, 25, 28, 16, 1, 6, 20, 44, 66, 64, 32, 1, 7, 27, 70, 129, 168, 144, 64, 1, 8, 35, 104, 225, 360, 416, 320, 128, 1, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 1, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512, 1, 11, 65

Offset: 1

N. J. A. Sloane, May 15 2010

Suggested by a question from Phyllis Chinn (Humboldt State University).
As triangle, mirror image of A105306. - Philippe Deléham, Nov 01 2011
A160232 is jointly generated with A208341 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + 2x*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Feb 25 2012
Subtriangle of the triangle T(n,k) given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012

			Array begins:
  1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, ...
  1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, ...
  1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, ...
  1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, ...
  1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, ...
From _Clark Kimberling_, Feb 25 2012: (Start)
As a triangle (see Comments):
  1;
  1,  1;
  1,  2,  2;
  1,  3,  5,  4;
  1,  4,  9, 12,  8;  (End)
From _Philippe Deléham_, Mar 08 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  2,  0;
  1,  3,  5,  4,  0;
  1,  4,  9, 12,  8,  0;
  1,  5, 14, 25, 28, 16,  0; (End)

Rows give A011782, A045623, A058396, A062109, A169792-A169797.

Cf. A062110, A105306, A208341.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208341 *)
(* Clark Kimberling, Feb 25 2012 *)

From Philippe Deléham, Mar 08 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-2*y*x)/(1-2*y*x-x+y*x^2).
Sum_{k=0..n, n>0} T(n,k)*x^k = A000012(n), A001519(n), A052984(n-1) for x = 0, 1, 2 respectively. (End)

A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 8, 12, 9, 4, 1, 0, 16, 28, 25, 14, 5, 1, 0, 32, 64, 66, 44, 20, 6, 1, 0, 64, 144, 168, 129, 70, 27, 7, 1, 0, 128, 320, 416, 360, 225, 104, 35, 8, 1, 0, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 0, 512, 1536, 2400, 2528, 1970

Offset: 0

Henry Bottomley, May 30 2001

The triangular version of this square array is defined by T(n,k) = A(k,n-k) for 0 <= k <= n. Conversely, A(n,k) = T(n+k,n) for n,k >= 0. We have [o.g.f of T](x,y) = [o.g.f. of A](x*y, x) and [o.g.f. of A](x,y) = [o.g.f. of T](y,x/y). - Petros Hadjicostas, Feb 11 2021
From Paul Barry, Nov 10 2008: (Start)
As number triangle, Riordan array (1, x(1-x)/(1-2x)). A062110*A007318 is A147703.
[0,1,1,0,0,0,....] DELTA [1,0,0,0,.....]. (Philippe Deléham's DELTA is defined in A084938.) (End)
Modulo 2, this triangle T becomes triangle A106344. - Philippe Deléham, Dec 18 2008

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1, 0,  0,   0,   0,    0,    0,     0,     0,     0, ...
  1, 1,  2,   4,   8,   16,   32,    64,   128,   256, ...
  1, 2,  5,  12,  28,   64,  144,   320,   704,  1536, ...
  1, 3,  9,  25,  66,  168,  416,  1008,  2400,  5632, ...
  1, 4, 14,  44, 129,  360,  968,  2528,  6448, 16128, ...
  1, 5, 20,  70, 225,  681, 1970,  5500, 14920, 39520, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, ...
  ... - _Petros Hadjicostas_, Feb 15 2021
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,   1;
  0,   4,   5,   3,   1;
  0,   8,  12,   9,   4,   1;
  0,  16,  28,  25,  14,   5,   1;
  0,  32,  64,  66,  44,  20,   6,   1;
  0,  64, 144, 168, 129,  70,  27,   7,   1;
  0, 128, 320, 416, 360, 225, 104,  35,   8,   1;
  ... - _Philippe Deléham_, Nov 30 2008

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, Upper Hessenberg and Toeplitz Bohemians, arXiv:1907.10677 [cs.SC], 2019.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq., 21 (2018), #18.1.4.

Rows of A include A000007, A011782, A045623, A058396, A062109, A169792, A169793, A169794, A169795, A169796, A169797.
Columns of A include A000012, A001477, A000096, A000297.
Main diagonal of A is A002002.
Table A(n, k) is a multiple of 2^(k-n); dividing by this gives a table similar to A050143 except at the edges.
Essentially the same array as A105306, A160232.

t[n_, n_] = 1; t[n_, k_] := 2^(n-2*k)*k*Hypergeometric2F1[1-k, n-k+1, 2, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Philippe Deléham + symbolic sum *)

a(i,j)=if(i<0 || j<0,0,polcoeff(((1-x)/(1-2*x)+x*O(x^j))^i,j))

Formulas for the square array (A(n,k): n,k >= 0):
A(n, k) = A(n-1, k) + Sum_{0 <= j < k} A(n, j) for n >= 1 and k >= 0 with A(0, k) = 0^k for k >= 0.
G.f.: 1/(1-x*(1-y)/(1-2*y)) = Sum_{i, j >= 0} A(i, j) x^i*y^j.
From Petros Hadjicostas, Feb 15 2021: (Start)
A(n,k) = 2^(k-n)*n*hypergeom([1-n, k+1], [2], -1) for n >= 0 and k >= 1.
A(n,k) = 2*A(n,k-1) + A(n-1,k) - A(n-1,k-1) for n,k >= 1 with A(n,0) = 1 for n >= 0 and A(0,k) = 0 for k >= 1. (End)
Formulas for the triangle (T(n,k): 0 <= k <= n):
From Philippe Deléham, Aug 01 2006: (Start)
T(n,k) = A121462(n+1,k+1)*2^(n-2*k) for 0 <= k < n.
T(n,k) = 2^(n-2*k)*k*hypergeom([1-k, n-k+1], [2], -1) for 0 <= k < n. (End)
Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Dec 09 2008
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 1 <= k <= n-1 with T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013
G.f.: Sum_{n.k>=0} T(n,k)*x^n*y^k = (1 - 2*x)/(x^2*y - x*y - 2*x + 1). - Petros Hadjicostas, Feb 15 2021

Various sections edited by Petros Hadjicostas, Feb 15 2021

A176479 a(n) = (n+1)*A001003(n).

Original entry on oeis.org

1, 2, 9, 44, 225, 1182, 6321, 34232, 187137, 1030490, 5707449, 31760676, 177435297, 994551222, 5590402785, 31500824304, 177880832001, 1006362234162, 5703029112297, 32367243171740, 183945502869345, 1046646207221582, 5961966567317649, 33995080211156904

Offset: 0

Paul Barry, Apr 18 2010

Central coefficients T(2n,n) of the Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)), A105306.
a(n) counts the bi-degree sequences of directed trees (i.e., digraphs whose underlying graph is a tree) with n edges. - Nikos Apostolakis, Dec 31 2016
a(n) is also the number of Dyck paths having exactly n peaks in level 1 and n peaks in level 2 and no other peaks. a(2) = 9: /\/\//\/\\, /\//\/\\/\, //\/\\/\/\, /\/\//\\//\\, /\//\\/\//\\, /\//\\//\\/\, //\\/\/\//\\, //\\/\//\\/\, //\\//\\/\/\. - Alois P. Heinz, Jun 20 2017
For n>0, a(n) is the number of ordered trees with n+1 leaves, no node of outdegree 1, and having one of its leaves marked. - Juan B. Gil, Jan 03 2024

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Juan B. Gil, Emma G. Hoover, and Jessica A. Shearer, Bijections between colored compositions, Dyck paths, and polygon partitions, arXiv:2403.04575 [math.CO], 2024.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3

Cf. A001003, A105306.

Row n=2 of A288972.

a:= proc(n) option remember; `if`(n<2, n+1,
     (6*n-3)/n*a(n-1) -(n-2)/(n-1)*a(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 22 2017

a[n_] := Sum[Binomial[n - 1, k - 1]*Binomial[n + k, n], {k, 0, n}]; Array[a, 25, 0] (* or *)
CoefficientList[ Series[1/4 - (x - 3)/(4 Sqrt[x^2 - 6x +1]), {x, 0, 25}], x] (* Robert G. Wilson v, Dec 31 2016 *)
Table[(n+1)Hypergeometric2F1[1-n, -n, 2, 2], {n,0,21}] (* Peter Luschny, Jan 02 2017 *)

E.g.f.: 1+exp(3*x)*Bessel_I(1,2*sqrt(2)*x)/sqrt(2) +int(exp(3*x) *Bessel_I(1,2*sqrt(2)*x) /(sqrt(2)*x),x).
G.f.: 1/4 - (x-3)/(4*sqrt(x^2-6*x+1)). - Dmitry Kruchinin, Aug 31 2012
Conjecture: n*(n-1)*a(n) -3*(2*n-1)*(n-1)*a(n-1) +n*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014
a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(n+k,n). - Nikos Apostolakis, Dec 31 2016
a(n) = (n+1)*hypergeom([1-n, -n], [2], 2). - Peter Luschny, Jan 02 2017

cp's OEIS Frontend

A195304 Decimal expansion of shortest length of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5).

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A077998 Expansion of (1-x)/(1-2*x-x^2+x^3).

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SageMath

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A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n.

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A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

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A176479 a(n) = (n+1)*A001003(n).

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Maple

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A106197 Analog of A094091 for S=4.

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A125692 Riordan array (1-x*c(-x^2),x(1-x*c(-x^2))) where c(x) is the g.f. of A000108.

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A125692 Riordan array (1-xc(-x^2),x(1-xc(-x^2))) where c(x) is the g.f. of A000108.