A120730 -id:A120730 - cp's OEIS frontend

A121725 Generalized central coefficients for k=3.

1, 1, 10, 19, 190, 442, 4420, 11395, 113950, 312814, 3128140, 8960878, 89608780, 264735892, 2647358920, 8006545891, 80065458910, 246643289830, 2466432898300, 7711583225338, 77115832253380, 244082045341036, 2440820453410360, 7805301802531534

Offset: 0

Paul Barry, Aug 17 2006

Hankel transform of a(n) is 9^binomial(n+1,2). Case k=3 of T(n,k) = (1/Pi)*2*k^2*(2*k)^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(1+k^2-2*k*x) dx. T(n,k) has Hankel transform (k^2)^binomial(n+1,2). k=1 corresponds to C(n, floor(n/2)).

Expansion of c(9*x^2)/(1-x*c(9*x^2)), where c(x) is the g.f. of A000108. Reversion of x*(1+x)/(1+2*x+10*x^2). - Philippe Deléham, Nov 09 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A009766, A120730.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-36*x^2))/(18*x^2-x*(1-Sqrt(1-36*x^2))) )); // G. C. Greubel, Nov 07 2022

CoefficientList[Series[(1-Sqrt[1-4*9*x^2])/(2*9*x^2-x*(1-Sqrt[1-4*9*x^2])), {x,0,40}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A121725(n): return sum(9^(n-k)*A120730(n,k) for k in range(n+1))
[A121725(n) for n in range(41)] # G. C. Greubel, Nov 07 2022

a(n) = (1/Pi)*18*6^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(10-6*x) dx.
a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^2k. - Philippe Deléham, Aug 18 2006
a(n) = Sum_{k=0..n} A120730(n,k)*9^(n-k). - Philippe Deléham, Nov 09 2007
Conjecture: (n+1)*a(n) = 10*(n+1)*a(n-1) + 36*(n-2)*a(n-2) - 360*(n-2)*a(n-3). - R. J. Mathar, Nov 26 2012
From Vaclav Kotesovec, Feb 13 2014: (Start)
a(n) ~ (4+(-1)^n) * 2^(n-7/2) * 3^(n+2) / (n^(3/2) * sqrt(Pi)).
G.f.: (1 - sqrt(1 - 36*x^2))/(18*x^2 - x*(1 - sqrt(1 - 36*x^2))). (End)

More terms from Vincenzo Librandi, Feb 15 2014

A156361 a(2n+2) = 7a(2n+1), a(2n+1) = 7a(2n) - 6^n*A000108(n), a(0) = 1.

Original entry on oeis.org

1, 6, 42, 288, 2016, 14040, 98280, 686880, 4808160, 33638976, 235472832, 1647983232, 11535882624, 80745019776, 565215138432, 3956385876480, 27694701135360, 193860506096640, 1357023542676480, 9499115800977408

Offset: 0

Philippe Deléham, Feb 08 2009

nonn

Hankel transform is 6^C(n+1, 2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.

Cf. A000108, A001405, A156128, A151162, A156195, A151254, A151281.

[n le 3 select Factorial(n+4)/120 else (7*n*Self(n-1) + 24*(n-3)*Self(n-2) - 168*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

A156361 := proc(n)
    option remember;
    local nh;
    if n= 0 then
        1;
    elif  type(n,'even') then
        7*procname(n-1);
    else
        nh := floor(n/2) ;
        7*procname(n-1)-6^nh*A000108(nh) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2016

a[n_]:= a[n]= If[n==0, 1, 7*a[n-1] -If[EvenQ[n], 0, 6^((n-1)/2)* CatalanNumber[(n-1)/2]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 04 2022 *)

def a(n): # a = A156361
    if (n==0): return 1
    elif (n%2==1): return 7*a(n-1) - 6^((n-1)/2)*catalan_number((n-1)/2)
    else: return 7*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum{k=0..n} A120730(n,k) * 6^k.

(n+1)*a(n) = 7*(n+1)*a(n-1) + 24*(n-2)*a(n-2) - 168*(n-2)*a(n-3). - R. J. Mathar, Jul 21 2016

A156362 a(2n+2) = 8a(2n+1), a(2n+1) = 8a(2n) - 7^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 7, 56, 441, 3528, 28126, 225008, 1798349, 14386792, 115060722, 920485776, 7363180314, 58905442512, 471228010428, 3769824083424, 30158239367445, 241265914939560, 1930119075851050, 15440952606808400, 123527424655229966

Offset: 0

Philippe Deléham, Feb 08 2009

nonn

Hankel transform is 7^C(n+1,2).

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

Cf. A000108, A001405, A151162, A156195, A151254, A151281, A156266, A156361.

[n le 3 select Factorial(n+5)/720 else (8*n*Self(n-1) + 28*(n-3)*Self(n-2) - 224*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 8*a[n-1] -7^((n-1)/2)*CatalanNumber[(n-1)/2], 8*a[n-1]]]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)

def a(n): # a = A156362
    if (n==0): return 1
    elif (n%2==1): return 8*a(n-1) - 7^((n-1)/2)*catalan_number((n-1)/2)
    else: return 8*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 7^k.

a(n) = ( 8*(n+1)*a(n-1) + 28*(n-2)*a(n-2) - 224*(n-2)*a(n-3) )/(n+1). - G. C. Greubel, Nov 09 2022

A165408 An aerated Catalan triangle.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 4, 0, 1, 0, 0, 0, 5, 0, 5, 0, 1, 0, 0, 0, 0, 9, 0, 6, 0, 1, 0, 0, 0, 5, 0, 14, 0, 7, 0, 1, 0, 0, 0, 0, 14, 0, 20, 0, 8, 0, 1, 0, 0, 0, 0, 0, 28, 0, 27, 0, 9, 0, 1, 0, 0, 0, 0, 14, 0, 48, 0, 35, 0, 10, 0, 1, 0, 0, 0, 0, 0, 42, 0, 75, 0, 44, 0, 11, 0, 1

Offset: 0

Paul Barry, Sep 17 2009

Aeration of A120730. Row sums are A165407.

T(n,k) is the number of lattice paths from (0,0) to (k,(n-k)/2) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}. - Alois P. Heinz, Sep 20 2022

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 2, 0,  1;
  0, 0, 0, 3,  0,  1;
  0, 0, 2, 0,  4,  0,  1;
  0, 0, 0, 5,  0,  5,  0,  1;
  0, 0, 0, 0,  9,  0,  6,  0,  1;
  0, 0, 0, 5,  0, 14,  0,  7,  0, 1;
  0, 0, 0, 0, 14,  0, 20,  0,  8, 0,  1;
  0, 0, 0, 0,  0, 28,  0, 27,  0, 9,  0, 1;
  0, 0, 0, 0, 14,  0, 48,  0, 35, 0, 10, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000108, A001405, A001477, A105523, A120730, A165407 (row sums), A165409.

Cf. A174687, A236194, A262394.

A165408:= func< n,k | n le 3*k select Binomial(Floor((n+k)/2), k)*((3*k-n)/2 +1)*(1+(-1)^(n-k))/(2*(k+1)) else 0 >;
[A165408(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Nov 09 2022

b:= proc(x, y) option remember; `if`(y<=x, `if`(x=0, 1,
      b(x-1, y)+`if`(y>0, b(x, y-1), 0)), 0)
    end:
T:= (n, k)-> `if`((n-k)::even, b(k, (n-k)/2), 0):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Sep 20 2022

b[x_, y_]:= b[x, y]= If[y<=x, If[x==0, 1, b[x-1, y] + If[y>0, b[x, y-1], 0]], 0];
T[n_, k_]:= If[EvenQ[n-k], b[k, (n-k)/2], 0];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* Jean-François Alcover, Oct 08 2022, after Alois P. Heinz *)

def A165408(n,k): return 0 if (n>3*k) else binomial(int((n+k)/2), k)*((3*k-n+2)/2 )*(1+(-1)^(n-k))/(2*(k+1))
flatten([[A165408(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Nov 09 2022

T(n,k) = if(n<=3k, C((n+k)/2, k)*((3*k-n)/2 + 1)*(1 + (-1)^(n-k))/(2*(k+1)), 0).
G.f.: 1/(1-x*y-x^3*y/(1-x^3*y/(1-x^3*y/(1-x^3*y/(1-... (continued fraction).
Sum_{k=0..n} T(n, k) = A165407(n).
From G. C. Greubel, Nov 09 2022: (Start)
Sum_{k=0..floor(n/2)} T(n-k, k) = (1+(-1)^n)*A001405(n/2)/2.
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1+(-1)^n)*A105523(n/2)/2.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A165407(n).
Sum_{k=0..n} 2^k*T(n, k) = A165409(n).
T(n, n-2) = A001477(n-2), n >= 2.
T(2*n, n) = (1+(-1)^n)*A174687(n/2)/2.
T(2*n, n+1) = (1-(-1)^n)*A262394(n/2)/2.
T(2*n, n-1) = (1+(-1)^n)*A236194(n/2)/2
T(3*n-2, n) = A000108(n), n >= 1. (End)

A132373 Expansion of c(6x^2)/(1-xc(6*x^2)), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 7, 13, 91, 205, 1435, 3565, 24955, 65821, 460747, 1265677, 8859739, 25066621, 175466347, 507709165, 3553964155, 10466643805, 73266506635, 218878998733, 1532152991131, 4631531585341, 32420721097387, 98980721277613, 692865048943291, 2133274258946845

Offset: 0

Philippe Deléham, Nov 10 2007

Hankel transform is 6^C(n+1, 2).

Series reversion of (1+x)/(1 + 2*x + 7*x^2). [Corrected by R. J. Mathar, Nov 19 2009]

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

Cf. A000012, A000108, A001405, A120730, A121724, A126087, A128386, A128387.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-24*x^2))/(12*x^2-x*(1-Sqrt(1-24*x^2))) )); // G. C. Greubel, Nov 07 2022

CoefficientList[Series[(1-Sqrt[1-24*x^2])/(12*x^2 -x*(1-Sqrt[1-24*x^2])), {x, 0, 40}], x] (* G. C. Greubel, Nov 07 2022 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A132373(n): return sum(6^(n-k)*A120730(n,k) for k in range(n+1))
[A132373(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 6^(n-k).
From G. C. Greubel, Nov 07 2022: (Start)
G.f.: (1 - sqrt(1-24*x^2))/(12*x^2 - x*(1 - sqrt(1-24*x^2))).
a(n) = ( 7*(n+1)*a(n-1) + 24*(n-2)*a(n-2) - 168*(n-2)*a(n-3) )/(n+1). (End)

Terms beyond a(7) added by R. J. Mathar, Nov 19 2009

A132375 Expansion of c(8x^2)/(1 - xc(8*x^2)), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 9, 17, 153, 353, 3177, 8113, 73017, 198401, 1785609, 5060433, 45543897, 133071009, 1197639081, 3581326065, 32231934585, 98156060225, 883404542025, 2730108129937, 24570973169433, 76862217117665, 691759954058985

Offset: 0

Philippe Deléham, Nov 10 2007

Hankel transform is 8^C(n+1, 2).

Series reversion of x*(1+x)/(1+2*x+9*x^2).

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

Cf. A001405, A126087, A128386, A121724, A128387, A121725.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-32*x^2))/(16*x^2 -x*(1-Sqrt(1-32*x^2))) )); // G. C. Greubel, Nov 08 2022

CoefficientList[Series[(1-Sqrt[1-32*x^2])/(16*x^2-x*(1-Sqrt[1-32*x^2])), {x,0, 40}], x] (* G. C. Greubel, Nov 08 2022 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A132375(n): return sum(8^(n-k)*A120730(n,k) for k in range(n+1))
[A132375(n) for n in range(51)] # G. C. Greubel, Nov 08 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 8^(n-k).
From G. C. Greubel, Nov 08 2022: (Start)
a(n) = (9*(n+1)*a(n-1) + 32*(n-2)*a(n-2) - 288*(n-2)*a(n-3))/(n+1).
G.f.: (1 - sqrt(1-32*x^2))/(16*x^2 - x*(1 - sqrt(1-32*x^2))). (End)

A156566 a(2n+2) = 9a(2n+1), a(2n+1) = 9a(2n) - 8^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 8, 72, 640, 5760, 51712, 465408, 4186112, 37675008, 339017728, 3051159552, 27459059712, 247131537408, 2224149233664, 20017343102976, 180155188248576, 1621396694237184, 14592546256715776, 131332916310441984

Offset: 0

Philippe Deléham, Feb 10 2009

nonn

Hankel transform is 8^C(n+1,2).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A001405, A151162, A151254, A151281.
Cf. A156195, A156270, A156361, A156362, A156577.
Cf. A001018, A120730.

a[0] = 1; a[1] = 8; a[2] = 72; a[n_] := a[n] = (-288*(n-2)*a[n-3] + 32*(n-2)*a[n-2] + 9*(n+1)*a[n-1])/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Nov 15 2016 *)
a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 9*a[n-1] - 8^((n-1)/2)*CatalanNumber[(n- 1)/2], 9*a[n-1]]]; Table[a[n], {n,0,30}] (* G. C. Greubel, May 18 2022 *)

def a(n): # a = A156566
    if (n==0): return 1
    elif (n%2==1): return 9*a(n-1) - 8^((n-1)/2)*catalan_number((n-1)/2)
    else: return 9*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, May 18 2022

a(n) = Sum_{k=0..n} A120730(n,k)*8^k.

A156577 a(2n+2) = 10a(2n+1), a(2n+1) = 10a(2n) - 9^n*A000108(n), a(0) = 1.

Original entry on oeis.org

1, 9, 90, 891, 8910, 88938, 889380, 8890155, 88901550, 888923646, 8889236460, 88889884542, 888898845420, 8888918303988, 88889183039880, 888889778505099, 8888897785050990, 88888916293698870, 888889162936988700

Offset: 0

Philippe Deléham, Feb 10 2009

nonn

Hankel transform is 9^binomial(n+1,2).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156195, A156273, A156361, A156362, A156566.

a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 10*a[n-1] -9^((n-1)/2)*CatalanNumber[(n-1)/2], 10*a[n-1] ]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 04 2022 *)

def a(n): # a = A156577
    if (n==0): return 1
    elif (n%2==1): return 10*a(n-1) - 9^((n-1)/2)*catalan_number((n-1)/2)
    else: return 10*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Jan 04 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 9^k.

A357654 Number of lattice paths from (0,0) to (i,n-2*i) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 6, 9, 10, 19, 29, 34, 63, 97, 118, 215, 333, 416, 749, 1165, 1485, 2650, 4135, 5355, 9490, 14845, 19473, 34318, 53791, 71313, 125104, 196417, 262735, 459152, 721887, 973027, 1694914, 2667941, 3619955, 6287896, 9907851, 13521307, 23429158

Offset: 0

Alois P. Heinz, Oct 07 2022

Alois P. Heinz, Table of n, a(n) for n = 0..2500
Wikipedia, Counting lattice paths

Cf. A120730, A165407, A357655.

A120730:= func< n, k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
A357654:= func< n | (&+[A120730(n-k, k): k in [0..Floor(n/2)]]) >;
[A357654(n): n in [0..50]]; // G. C. Greubel, Nov 07 2022

b:= proc(x, y) option remember; `if`(min(x, y)<0 or y>x, 0,
      `if`(max(x, y)=0, 1, b(x-1, y)+b(x, y-1)))
    end:
a:= n-> add(b(i, n-2*i), i=ceil(n/3)..floor(n/2)):
seq(a(n), n=0..44);

A120730[n_, k_]:= If[n>2*k, 0, Binomial[n,k]*(2*k-n+1)/(k+1)];
A357654[n_]:= Sum[A120730[n-k,k], {k,0,Floor[n/2]}];
Table[A357654[n], {n,0,50}] (* G. C. Greubel, Nov 07 2022 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A357654(n): return sum(A120730(n-k,k) for k in range((n//2)+1))
[A357654(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

a(n) = Sum_{k=0..floor(n/2)} A120730(n-k, k). - G. C. Greubel, Nov 07 2022

A357825 Total number of n-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j = 0..floor(n/2).

Original entry on oeis.org

1, 1, 2, 9, 98, 4150, 562692, 211106945, 404883552194, 1766902576146876, 40519034229909243476, 2708397617879598970178238, 658332084097982587522119612196, 735037057881394837614680080889845116, 2030001034486747324990010196845670569155080

Offset: 0

Alois P. Heinz, Oct 14 2022

Alois P. Heinz, Table of n, a(n) for n = 0..60
Vaclav Kotesovec, Graph - the asymptotic ratio (5000 terms)
Wikipedia, Counting lattice paths

Main diagonal of A357824.

Cf. A000108, A000225, A008315, A120730, A357871.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> add(b(n, n-2*j)^n, j=0..n/2):
seq(a(n), n=0..15);

Table[Sum[(Binomial[n, k]*(n - 2*k + 1)/(n - k + 1))^n, {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 17 2022 *)

a(n) = A357824(n,n).
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^n.
a(n) = Sum_{j=0..n} A120730(n,j)^n.
a(n) mod 2 = 1 <=> n in { A000225 }.
From Vaclav Kotesovec, Nov 17 2022: (Start)
a(n)^(1/n) ~ exp(-1/2) * 2^(n + 3/2) / (sqrt(Pi)*n).
Limit_{n->oo} a(n) / (2^(n^2 + 3*n/2) / (n^n * exp(n/2) * Pi^(n/2))) does not exist, see also graph. (End)
Conjecture: the superconguence a(2*p-1) == 1 (mod p^3) holds for all primes p >= 5 (checked up to p = 101). - Peter Bala, Mar 20 2023

cp's OEIS Frontend

A121725 Generalized central coefficients for k=3.

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A156361 a(2*n+2) = 7*a(2*n+1), a(2*n+1) = 7*a(2*n) - 6^n*A000108(n), a(0) = 1.

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A156362 a(2*n+2) = 8*a(2*n+1), a(2*n+1) = 8*a(2*n) - 7^n*A000108(n), a(0)=1.

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A165408 An aerated Catalan triangle.

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A132373 Expansion of c(6*x^2)/(1-x*c(6*x^2)), where c(x) is the g.f. of A000108.

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A132375 Expansion of c(8*x^2)/(1 - x*c(8*x^2)), where c(x) is the g.f. of A000108.

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A156566 a(2n+2) = 9*a(2n+1), a(2n+1) = 9*a(2n) - 8^n*A000108(n), a(0)=1.

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A156361 a(2n+2) = 7a(2n+1), a(2n+1) = 7a(2n) - 6^n*A000108(n), a(0) = 1.

A156362 a(2n+2) = 8a(2n+1), a(2n+1) = 8a(2n) - 7^n*A000108(n), a(0)=1.

A132373 Expansion of c(6x^2)/(1-xc(6*x^2)), where c(x) is the g.f. of A000108.

A132375 Expansion of c(8x^2)/(1 - xc(8*x^2)), where c(x) is the g.f. of A000108.

A156566 a(2n+2) = 9a(2n+1), a(2n+1) = 9a(2n) - 8^n*A000108(n), a(0)=1.

A156577 a(2n+2) = 10a(2n+1), a(2n+1) = 10a(2n) - 9^n*A000108(n), a(0) = 1.