A037027 -id:A037027 - cp's OEIS frontend

A213684 Logarithmic derivative of A001002.

1, 5, 22, 105, 511, 2534, 12720, 64449, 328900, 1688115, 8705060, 45064110, 234054198, 1219053680, 6364813192, 33302104593, 174570695175, 916628799380, 4820160541350, 25381091113455, 133808636072595, 706211862466500, 3730964595817680, 19729042153581150

Offset: 1

Paul D. Hanna, Jun 22 2012

A001002(n) is the number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.
The g.f. of A001002 satisfies: G(x) = 1 + x*G(x)^2 + x^2*G(x)^3.
Central terms in A155161: a(n) = A155161(2*n,n). - Reinhard Zumkeller, Apr 17 2013
a(n) is the 2n-th term of the n-fold self-convolution of the Fibonacci numbers. - Alois P. Heinz, Feb 07 2021

			L.g.f.: L(x) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 511*x^5/5 +...
such that
L(x) = x*(1+x) + d/dx x^3*(1+x)^2/2! + d^2/dx^2 x^5*(1+x)^3/3! + d^3/dx^3 x^7*(1+x)^4/4! +...
The g.f. of A001002 begins:
exp(L(x)) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 154*x^5 + 654*x^6 +...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Dmitry V. Kruchinin and Vladimir V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.

Cf. A000045, A001002, A037027, A155161, A298610, A350383.

a213684 n = a155161 (2*n) n  -- Reinhard Zumkeller, Apr 17 2013

with(orthopoly): seq(add(i, i in [seq((-1)^iquo(n-k,2)*coeff(G(n,n,x/2), x, k), k=0..n)]), n=1..24); # Peter Luschny, Jan 26 2018

Table[n*Sum[Binomial[k+n-1,n]*Binomial[k,n-k]/k,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=n*sum(r=1,n,binomial(r+n-1,n)*binomial(r,n-r)/r)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=(sum(m=1, n+1, Dx(m-1, x^(2*m-1)*(1+x)^m/m!)+x*O(x^n))); n*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

a(n) = n * Sum_{r=1..n} binomial(r+n-1,n) * binomial(r,n-r) / r.
L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1+x)^n/n! = Sum_{n>=1} a(n)*x^n/n.
Recurrence: 75*(n-1)*n*a(n) = 5*(n-1)*(59*n-12)*a(n-1) + (559*n^2-1503*n+1100)* a(n-2) + 21*(3*n-8)*(3*n-7)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3^(3*n)/(2*5^(n-1/2)*sqrt(6*Pi*n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = A037027(2*n-1,n-1). - Vladimir Kruchinin, Feb 28 2013
a(n) = Sum_{k=0..n} (-1)^floor((n-k)/2) [x^k] G(n,n,x/2), where G(n,a,x) denotes the n-th Gegenbauer polynomial; row sums of A298610. - Peter Luschny, Jan 26 2018
a(n) = [x^n] (1/(1-x-x^2))^n. - Alois P. Heinz, Feb 07 2021
From Peter Bala, Mar 11 2025: (Start)
a(n) = Sum_{k = 0..n} n/(2*n-k) * binomial(2*n-k, k)*binomial(2*n-2*k, n).
a(n) = (1/2)*binomial(2*n, n)*hypergeom([-n/2, (-n+1)/2], [-2*n+1], -4). Cf. A350383.
Second-order recurrence: 5*n*(n-1)*(8*n-13)*a(n) = 2*(n-1)*(88*n^2-187*n+75)*a(n-1) + 3*(8*n-5)*(3*n-4)*(3*n-5)*a(n-2) with a(1) = 1 and a(2) = 5. (End)

A122075 Coefficients of a generalized Pell-Lucas polynomial read by rows.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 5, 7, 4, 1, 8, 15, 12, 5, 1, 13, 30, 31, 18, 6, 1, 21, 58, 73, 54, 25, 7, 1, 34, 109, 162, 145, 85, 33, 8, 1, 55, 201, 344, 361, 255, 125, 42, 9, 1, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 144, 655, 1416, 1918, 1806, 1239, 630, 236, 63, 11, 1

Offset: 0

R. J. Mathar, Oct 16 2006

A122075 is jointly generated with A037027 as an 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+1)*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012

Subtriangle of the triangle T(n,k) given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 11 2012

			Triangle begins:
  1
  2 1
  3 3 1
  5 7 4 1
  8 15 12 5 1
  13 30 31 18 6 1
A055830 = (1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, ...) begins:
  1
  1, 0
  2, 1, 0
  3, 3, 1, 0
  5, 7, 4, 1, 0
  8, 15, 12, 5, 1, 0
  13, 30, 31, 18, 6, 1, 0

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Tian-Xiao He and Peter J.-S. Shiue, Identities for linear recursive sequences of order 2, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
Y. Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370.

See A055830 for another version.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 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[%]    (* A122075 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A037027 *)
(* Clark Kimberling, Mar 05 2012 *)
CoefficientList[CoefficientList[Series[-(1 + x)/(-1 + x*y + x + x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Dec 24 2017 *)

T(n,k)={ sum(j=0,n-k+1, binomial(n-k-j+1,j)*binomial(n-j,k)) ; } { nmax=10 ; for(n=0,nmax, for(k=0,n, print1(T(n,k),",") ; ); ); }

T(n,k) = Sum_{j=0..n-k+1} binomial(n-k-j+1,j)*binomial(n-j,k).
Sum_{k>=0} T(n-k,k) = 2^n.
Sum_{k>=0} (-1)^k*T(n-k,k) = 2-delta(0,n).
G.f.: -(1+x)/(-1+x*y+x+x^2). - R. J. Mathar, Aug 11 2015

A057280 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057995.

Original entry on oeis.org

2, 17, 5, 225, 120, 15, 4080, 3050, 700, 50, 94440, 89225, 28625, 3775, 175, 2666880, 3006000, 1208975, 223175, 19225, 625, 89016480, 115299900, 54824650, 12689800, 1537100, 93500, 2250, 3430929600, 4973077800, 2695596850, 737744125

Offset: 0

Wolfdieter Lang, Sep 13 2000

The row polynomials are q(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) =( p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057995(k,m).

			k=2: F2(n)=((16+5*n)*(n+1)*F0(n+1)+(17+5*n)*(n+2)*F0(n))/50, cf. A001628.

Cf. A000045, A037027, A057995.

A057995 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057280.

Original entry on oeis.org

1, 16, 5, 300, 160, 20, 6840, 4850, 1075, 75, 186120, 159650, 48175, 6100, 275, 5916240, 5846700, 2168650, 379700, 31550, 1000, 215717040, 238437900, 103057800, 22426825, 2605175, 153875, 3625, 8888140800, 10772348400

Offset: 0

Wolfdieter Lang, Sep 13 2000

The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of F0(n) := A000045(n+1) n >= 0, (Fibonacci starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057280).

			k=2: F2(n)=((16+5*n)*(n+1)*F0(n+1)+(17+5*n)*(n+2)*F0(n))/50, cf. A001628.

Cf. A000045, A037027, A057280.

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 5, 20, 30, 20, 5, 10, 30, 30, 10, 10, 20, 10, 5, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1

Offset: 0

Harlan J. Brothers, Jun 16 2010

P_h = level h of Pascal's prism where P_1 = Pascal's triangle (A007318) and P_2 = denominators of Leibniz harmonic triangle (A003506). A sequence of length k through P is defined by P for n = {1, 2, 3, ..., k}.

			Prism begins (levels 1-4):
1
1 1
1 2 1
1 3 3 1
1
2 2
3 6 3
4 12 12 4
1
3 3
6 12 6
10 30 30 10
1
4 4
10 20 10
20 60 60 20

H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Level 1 = A007318.
Level 2 = A003506.
Level 3 = A094305.
Level 4 = A178820.
Level 5 = A178821.
Level 6 = A178822.
Sums of shallow diagonals for each level correspond to rows of square A037027.
Contains A109649 and A046816.
P = A000984.
P = A006480.
P = A000897.
P<3n-2, 3n-2, n> = A113424.

end = 5; Column/@Table[Multinomial[h, i-j, j], {h, 0, end}, {i, 0, end}, {j, 0, i}]

a_(h, i, j) = (h+i-2; h-1, i-j, j-1), h >= 1, i >= 1, 1 <= j <= i.
Recurrence:
For P_h, element a is given by: a_(1, 1) = 1; a_(i, j) = ((i+h-2)/(i-1)) (a_(i-1, j) + a_(i-1, j-1)).

Keyword tabf by Michel Marcus, Oct 22 2017

A214178 Triangle T(n,k) by rows: the k-th derivative of the Fibonacci Polynomial F_n(x) evaluated at x=1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 2, 2, 0, 3, 5, 6, 6, 0, 5, 10, 18, 24, 24, 0, 8, 20, 44, 84, 120, 120, 0, 13, 38, 102, 240, 480, 720, 720, 0, 21, 71, 222, 630, 1560, 3240, 5040, 5040, 0, 34, 130, 466, 1536, 4560, 11760, 25200, 40320, 40320, 0, 55, 235, 948, 3564, 12264

Offset: 0

R. J. Mathar, L. Edson Jeffery, Reinhard Zumkeller, Jul 07 2012

T(n,0) = A000045(n), Fibonacci numbers;
T(n,1) = A001629(n) for n > 0;
T(n,n-3) = A038720(n-2) for n > 2;
T(n,n-2) = A000142(n-1) for n > 1;
T(n,n-1) = A000142(n-1) for n > 0;
T(n,n) = 0.

			The triangle begins:
.   0: [0]
.   1: [1, 0]
.   2: [1, 1, 0]
.   3: [2, 2, 2, 0]
.   4: [3, 5, 6, 6, 0]
.   5: [5, 10, 18, 24, 24, 0]
.   6: [8, 20, 44, 84, 120, 120, 0]
.   7: [13, 38, 102, 240, 480, 720, 720, 0]
.   8: [21, 71, 222, 630, 1560, 3240, 5040, 5040, 0]
.   9: [34, 130, 466, 1536, 4560, 11760, 25200, 40320, 40320, 0]
.  10: [55, 235, 948, 3564, 12264, 37800, 100800, 221760, 362880, 362880, 0]
       ...

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
P. Filipponi, A. F. Horadam, Second derivative sequences of Fibonacci and Lucas Polynomials, Fib. Quart. 31 (1993), 194-204.

Cf. A000142, A037027.

a214178 n k = a214178_tabl !! n !! k
a214178_row n = a214178_tabl !! n
a214178_tabl = [0] : map f a037027_tabl where
   f row = (zipWith (*) a000142_list row) ++ [0]

T[n_, k_] := D[Fibonacci[n, x], {x, k}] /. x -> 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 20 2021 *)

T(n,k) = A037027(n,k)*k!, 0 <= k < n; T(n,n) = 0.

A057282 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057281.

Original entry on oeis.org

2, 5, 17, 15, 120, 225, 50, 700, 3050, 4080, 175, 3775, 28625, 89225, 94440, 625, 19225, 223175, 1208975, 3006000, 2666880, 2250, 93500, 1537100, 12689800, 54824650, 115299900, 89016480, 8125, 438250, 9670750, 112454500, 737744125

Offset: 1

Wolfdieter Lang, Sep 13 2000

The row polynomials are q(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A057281(k,m).
a(k,0)= A020876(k), k >= 0.

			k=2: F2(n)=((5*n^2+21*n+16)*F(n+2)+(5*n^2+27*n+34)*F(n+1))/50, F(n) := A000045(n); see A001628.
2; 5,17; 15,120,225; 50,700,3050,4080; 175,3775,28625,89225,94440; ...

Cf. A000045, A037027, A057995, A057280, A057281.

Row sums: A151615.

A092565 Triangle of coefficients T(n,k) (n>=0, 0<=k<=2*n), read by rows, where the n-th row polynomial equals the numerator of the n-th convergent of the continued fraction [1+x+x^2;1+x+x^2,1+x+x^2,...] for n>0, with the zeroth row defined as T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 2, 1, 3, 5, 8, 7, 6, 3, 1, 5, 10, 19, 22, 22, 16, 10, 4, 1, 8, 20, 42, 58, 69, 63, 49, 30, 15, 5, 1, 13, 38, 89, 142, 191, 206, 191, 146, 95, 50, 21, 6, 1, 21, 71, 182, 327, 491, 602, 637, 573, 447, 296, 167, 77, 28, 7, 1, 34, 130, 363, 722, 1191, 1626

Offset: 0

Paul D. Hanna, Feb 28 2004

T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0),(2,0),(1,1),(1,2). - Joerg Arndt, Jul 01 2011

Diagonal forms A092566, row sums form A006190. Column T(n,0) forms Fibonacci numbers A000045, T(n,1) forms A001629.

			Ratio of row polynomials R(3)/R(2) = (3+5*x+8*x^2+7*x^3+6*x^4+3*x^5+x^6)/(2+2*x+3*x^2+2*x^3+x^4) = [1+x+x^2;1+x+x^2,1+x+x^2].
Rows begin:
  1;
  1, 1, 1;
  2, 2, 3, 2, 1;
  3, 5, 8, 7, 6, 3, 1;
  5, 10, 19, 22, 22, 16, 10, 4, 1;
  8, 20, 42, 58, 69, 63, 49, 30, 15, 5, 1;
  13, 38, 89, 142, 191, 206, 191, 146, 95, 50, 21, 6, 1;
  21, 71, 182, 327, 491, 602, 637, 573, 447, 296, 167, 77, 28, 7, 1;
  34, 130, 363, 722, 1191, 1626, 1921, 1958, 1752, 1366, 931, 546, 273, 112, 36, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..100, flattened

Cf. A092566, A037027.

T:= proc(x, y) option remember; `if`(y<0 or y>2*x, 0, `if`(x=0, 1,
      add(T(x-l[1], y-l[2]), l=[[1, 0], [2, 0], [1, 1], [1, 2]])))
    end:
seq(seq(T(n,k), k=0..2*n), n=0..10); # Alois P. Heinz, Apr 16 2013

A037027[n_, k_] := Sum[Binomial[k+j, k]*Binomial[j, n-j-k], {j, 0, n-k}]; A037027[n_, 0] = Fibonacci[n + 1]; row[n_] := CoefficientList[ Sum[A037027[n, k] x^k (1+x)^k, {k, 0, n}], x]; Flatten[Table[row[n], {n, 0, 8}]][[1 ;; 70]] (* Jean-François Alcover, Jul 18 2011 *)

PARI
```
T(n,k)=if(2*n
				
```

/* same as in A092566, but last line (output) replaced by the following */
/* show as triangle (0<=k<=2*n): */
{for (n=1,N, for (k=1,2*n-1, print1(M[n,k],", "); ); print(); );}
/* Joerg Arndt, Jul 01 2011 */

n-th row polynomial R(n) = Sum_{k=0..n} A037027(n, k)*x^k*(1+x)^k; R(n+1)/R(n) = [1+x+x^2;1+x+x^2, ...(n+1)times..., 1+x+x^2] for n>=0; R(0)=1.

A152440 Riordan matrix (1/(1-x-x^2),x/(1-x-x^2)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 9, 5, 1, 5, 22, 20, 7, 1, 8, 51, 65, 35, 9, 1, 13, 111, 190, 140, 54, 11, 1, 21, 233, 511, 490, 255, 77, 13, 1, 34, 474, 1295, 1554, 1035, 418, 104, 15, 1, 55, 942, 3130, 4578, 3762, 1925, 637, 135, 17, 1, 89, 1836, 7285, 12720, 12573, 7865, 3276

Offset: 0

Emanuele Munarini, Dec 04 2008, Dec 05 2008

From Philippe Deléham, Feb 20 2014: (Start)
T(n,0)   = A000045(n+1);
T(n+1,1) = A001628(n);
T(n+2,2) = A001873(n);
T(n+3,3) = A001875(n).
Row sums are A238236(n). (End)

			Triangle begins:
1;
1, 1;
2, 3, 1;
3, 9, 5, 1;
5, 22, 20, 7, 1;
8, 51, 65, 35, 9, 1;
13, 111, 190, 140, 54, 11, 1;
21, 233, 511, 490, 255, 77, 13, 1, etc.
- _Philippe Deléham_, Feb 20 2014

The first row is given by A000045.

Cf. A037027, A238241.

a(n,k) = sum( binomial(n-j-k,2k) binomial(n-j-k,j), j=0...(n-k)/2 )
a(n,k) = sum( binomial(i+2k,2k) binomial(n-i+k,i+2k), i=0...(n - k)/2 )
Recurrence: a(n+4,k+1) - 2 a(n+3,k+1) - a(n+3,k) - a(n+2,k+1) + 2 a(n+1,k+1) + a(n,k+1) = 0
GF for columns: 1/(1-x-x^2)(x/(1-x-x^2)^2)^k
GF: (1-x-x^2)/((1-x-x^2)^2-xy)
T(n,k) = A037027(n+k, 2*k). - Philippe Deléham, Feb 20 2014

A184018 Expansion of c(x/(1-x-x^2)) / (1-x-x^2), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 6, 19, 67, 254, 1017, 4236, 18168, 79680, 355635, 1609912, 7373401, 34102976, 159055728, 747211753, 3532452169, 16792693562, 80224098381, 384948157635, 1854469572120, 8965866981294, 43488834409737, 211569299607282

Offset: 0

Paul Barry, Jan 08 2011

Hankel transform is (9,-5) Somos-4 sequence A184019.

The radius of convergence r of the g.f. A(x) satisfies: r = (1-r-r^2)/4 = limit a(n)/a(n+1) = (sqrt(29)-5)/2 = 0.19258240... with A(r) = 1/(2*r) = (sqrt(29)+5)/4 = 2.59629120... - Paul D. Hanna, Sep 06 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A037027 := proc(n,m) add( binomial(m+k,m)*binomial(k,n-k-m),k=0..n-m) ; end proc:
A184018 := proc(n) add( A037027(n,k)*A000108(k),k=0..n) ; end proc:
seq(A184018(n),n=0..10) ; # R. J. Mathar, Jan 11 2011

CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 6 x^3 + x^4])/(2 x (1 - x - x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

{a(n)=polcoeff((1-sqrt(1-4*x/(1-x-x^2 +O(x^(n+2)))))/(2*x),n)} /* Paul D. Hanna, Sep 06 2011 */

G.f.: ( 1 - x - x^2 - sqrt((1 - x - x^2)*(1 - 5*x - x^2)) )/( 2*x*(1 - x - x^2) ).
G.f.: 1/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1-... (continued fraction).
a(n) = Sum_{k=0..n} (Sum_{j=0..n-k} binomial(k+j,k)*binomial(j,n-k-j))*A000108(k) = Sum_{k=0..n} A037027(n,k)*A000108(k).
G.f. satisfies A(x) = 1/(1-x-x^2) + x*A(x)^2. - Paul D. Hanna, Sep 06 2011
Conjecture: (n+1)*a(n) + 2*(1-3*n)*a(n-1) + 3*(n-1)*a(n-2) + 2*(3*n-5)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011
a(n) ~ (27+5*sqrt(29)) * sqrt(54*sqrt(29)-290) * (5+sqrt(29))^n / (sqrt(Pi) * n^(3/2) * 2^(n+5)). - Vaclav Kotesovec, Feb 04 2014

cp's OEIS Frontend

A213684 Logarithmic derivative of A001002.

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A122075 Coefficients of a generalized Pell-Lucas polynomial read by rows.

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A057280 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057995.

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A057995 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057280.

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A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

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A214178 Triangle T(n,k) by rows: the k-th derivative of the Fibonacci Polynomial F_n(x) evaluated at x=1.

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A057282 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057281.

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A092565 Triangle of coefficients T(n,k) (n>=0, 0<=k<=2*n), read by rows, where the n-th row polynomial equals the numerator of the n-th convergent of the continued fraction [1+x+x^2;1+x+x^2,1+x+x^2,...] for n>0, with the zeroth row defined as T(0,0)=1.

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