A026616 -id:A026616 - cp's OEIS frontend

A026622 a(n) = Sum_{k=0..n} A026615(n, k).

1, 2, 5, 12, 26, 54, 110, 222, 446, 894, 1790, 3582, 7166, 14334, 28670, 57342, 114686, 229374, 458750, 917502, 1835006, 3670014, 7340030, 14680062, 29360126, 58720254, 117440510, 234881022, 469762046, 939524094, 1879048190, 3758096382, 7516192766

Offset: 0

Clark Kimberling

In general, a first order inhomogeneous recurrence of the form s(0) = a, s(n) = m*s(n-1) + k, n>0, will have a closed form of a*m^n +((m^n-1)/(m-1))*k. - Gary Detlefs, Jun 07 2024

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026623, A026624, A026625, A026956, A026957, A026958, A026959.
Cf. A026960.

[n le 1 select n+1 else 7*2^(n-2) -2: n in [0..40]]; // G. C. Greubel, Jun 24 2024

Table[7*2^(n-2) -2 +Boole[n==1]/2 +(5/4)*Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 24 2024 *)

Vec((1-x+x^2+x^3)/((1-x)*(1-2*x)) + O(x^40)) \\ Colin Barker, Feb 17 2016

[(7*2^n -8 +2*int(n==1) +5*int(n==0))/4 for n in range(41)] # G. C. Greubel, Jun 24 2024

a(n) = 7 * 2^(n-2) - 2, a(0) = 1, a(1) = 2 (Cf. A026624). - Ralf Stephan, Feb 05 2004
a(n) = 2*a(n-1) + 2, n>2. - Gary Detlefs, Jun 22 2010
From Colin Barker, Feb 17 2016: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
G.f.: (1 - x + x^2 + x^3)/((1 - x)*(1 - 2*x)). (End)
E.g.f.: (1/4)*( 5 + 2*x - 8*exp(x) + 7*exp(2*x) ). - G. C. Greubel, Jun 24 2024

A026615 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = 2*n-1 for n >= 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2 and n >= 4.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 10, 7, 1, 1, 9, 17, 17, 9, 1, 1, 11, 26, 34, 26, 11, 1, 1, 13, 37, 60, 60, 37, 13, 1, 1, 15, 50, 97, 120, 97, 50, 15, 1, 1, 17, 65, 147, 217, 217, 147, 65, 17, 1, 1, 19, 82, 212, 364, 434, 364, 212, 82, 19, 1

Offset: 0

Clark Kimberling

T(n, k) is the number of paths from (0, 0) to (n-k, k) in the directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i=0, j >= 0 and for j=0, i >= 0.

			Triangle begins as:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  5,   1;
  1,  7, 10,   7,   1;
  1,  9, 17,  17,   9,   1;
  1, 11, 26,  34,  26,  11,   1;
  1, 13, 37,  60,  60,  37,  13,   1;
  1, 15, 50,  97, 120,  97,  50,  15,  1;
  1, 17, 65, 147, 217, 217, 147,  65, 17,  1;
  1, 19, 82, 212, 364, 434, 364, 212, 82, 19,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A026616, A026617, A026618, A026619, A026620, A026621, A026623.
Cf. A026624, A026956, A026957, A026958, A026959, A026960.
Cf. A000012, A005408, A098749, A145066, A176742.
Cf. A026622 (row sums), A026625 (diagonal sums).

function T(n,k) // T = A026615
   if k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return 2*n-1;
   else return T(n-1, k-1) + T(n-1, k);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2024

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, 2*n-1, T[n-1, k -1] + T[n-1,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2024 *)

def T(n,k): # T = A026615
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return 2*n-1
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 13 2024

Sum_{k=0..n} T(n, k) = A026622(n) (row sums).
From G. C. Greubel, Jun 13 2024: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000012(n).
T(n, 1) = A005408(n-1), n >= 1.
T(n, 2) = A098749(n), n >= 2.
T(n, 3) = A145066(n-2) - [n=3], n >= 3.
Sum_{k=0..n} (-1)^k*T(n, k) = A176742(n) + [n=2].
Sum_{k=0..n} (-1)^k*T(n-k, k) = b(n-2) + 2*[n=0] + [n=1], where b(n) = (1/6)*(-2*sqrt(3)*sin(Pi*n/3) + 2*sqrt(3)*sin(5*Pi*n/3) + 3*cos(Pi* n/2) + 3*cos(3*Pi*n/2) - 6).
Sum_{k=0..n} k*T(n, k) = n*(7*2^(n-3) - 1) + (1/4)*[n=1]. (End)

Offset corrected by G. C. Greubel, Jun 13 2024

A026617 a(n) = A026615(2*n, n-1).

Original entry on oeis.org

1, 7, 26, 97, 364, 1374, 5214, 19877, 76076, 292162, 1125332, 4345642, 16819256, 65226620, 253403190, 986022765, 3842200140, 14991031770, 58558504620, 228986816190, 896300806440, 3511441192740

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A026615, A026616, A026618, A026619, A026620, A026621, A026622.
Cf. A026623, A026624, A026625, A026956, A026957, A026958, A026959.
Cf. A026960, A176742.

[n eq 1 select 1 else (7*n^2-4*n+1)*Binomial(2*n-3, n-2)/Binomial(n+1, 2): n in [1..40]]; // G. C. Greubel, Jun 13 2024

Table[(7*n^2-4*n+1)*Binomial[2*n-3, n-2]/Binomial[n+1,2] - 3*Boole[n== 1], {n, 40}] (* G. C. Greubel, Jun 13 2024 *)

[(7*n^2-4*n+1)*binomial(2*n-3, n-2)/binomial(n+1, 2) - 3*int(n==1) for n in range(1,41)] # G. C. Greubel, Jun 13 2024

From G. C. Greubel, Jun 13 2024: (Start)
a(n) = (7*n^2 - 4*n + 1)*binomial(2*n-3, n-2)/binomial(n+1, 2) - 3*[n= 1].
G.f.: ( 2 - 5*x + 2*x^2 - (2 - x + 2*x^2)*sqrt(1 - 4*x) )/(2*x*sqrt(1-4*x)).
E.g.f.: (1/2)*( (1 - 2*x) - (1 - 2*x)*exp(2*x)*BesselI(0, 2*x) + 2*(2 - x)*exp(2*x)*BesselI(1, 2*x) ). (End)

A026618 a(n) = A026618(2*n, n-2).

Original entry on oeis.org

1, 11, 50, 212, 870, 3509, 14014, 55640, 220116, 868870, 3425092, 13491064, 53117350, 209097945, 823111350, 3240499440, 12759776700, 50254414650, 197979380220, 780170359800, 3075303389340

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026615, A026616, A026617, A026619, A026620, A026621, A026622.
Cf. A026623, A026624, A026625, A026956, A026957, A026958, A026959.
Cf. A026960.

[n eq 2 select 1 else (7*n^2-4*n+4)*Binomial(2*n,n-2)/(2*Binomial(2*n,2)): n in [2..40]]; // G. C. Greubel, Jun 13 2024

Table[(7*n^2-4*n+4)*Binomial[2*n,n-2]/(2*Binomial[2*n,2]) -Boole[n==2], {n,2,40}] (* G. C. Greubel, Jun 13 2024 *)

[(7*n^2-4*n+4)*binomial(2*n, n-2)/(2*binomial(2*n, 2)) - int(n==2) for n in range(2,41)] # G. C. Greubel, Jun 13 2024

From G. C. Greubel, Jun 13 2024: (Start)
a(n) = (7*n^2 - 4*n + 4)*binomial(2*n, n-2)/(2*binomial(2*n, 2)) -[n=2].
G.f.: ( (2 - 9*x + 8*x^2 - 2*x^3) - (2 - 5*x + 2*x^2 + 2*x^4)*sqrt(1 - 4*x) )/(2*x^2*sqrt(1-4*x)).
E.g.f.: exp(2*x)*( (3 - x)*BesselI(0, 2*x) + (x - 1 - 2/x)*BesselI(1, 2*x) ) - (1 + x^2/2). (End)

A026619 a(n) = A026615(2*n-1, n-1).

Original entry on oeis.org

1, 5, 17, 60, 217, 798, 2970, 11154, 42185, 160446, 613054, 2351440, 9048522, 34916300, 135059220, 523521630, 2033066025, 7908332190, 30807696150, 120173896920, 469334610030, 1834970026500

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A026615, A026616, A026617, A026618, A026620, A026621, A026622.
Cf. A026623, A026624, A026625, A026956, A026957, A026958, A026959.
Cf. A026960.

[n eq 1 select 1 else (7*n-4)*(n+1)*Catalan(n)/(4*(2*n-1)): n in [1..40]]; // G. C. Greubel, Jun 13 2024

Table[(7*n-4)*Binomial[2*n,n]/(4*(2*n-1)) -(1/2)*Boole[n==1], {n,40}] (* G. C. Greubel, Jun 13 2024 *)

[(7*n-4)*binomial(2*n,n)/(4*(2*n-1)) -(1/2)*int(n==1) for n in range(1,41)] # G. C. Greubel, Jun 13 2024

From G. C. Greubel, Jun 13 2024: (Start)
a(n) = (7*n - 4)*binomial(2*n, n)/(4*(2*n-1)) -(1/2)*[n=1].
G.f.: ( (2 - x) - (2 + x)*sqrt(1-4*x) )/(2*sqrt(1-4*x))
E.g.f.: (1/2)*exp(2*x)*( (2 - x)*BesselI(0, 2*x) + x*BesselI(1, 2*x) ) - (1 + x/2). (End)

A026620 a(n) = A026615(2*n-1, n-2).

Original entry on oeis.org

1, 9, 37, 147, 576, 2244, 8723, 33891, 131716, 512278, 1994202, 7770734, 30310320, 118343970, 462501135, 1809134115, 7082699580, 27750808470, 108812919270, 426966196410, 1676471166240, 6586744582080, 25894139638302, 101852815940622, 400840469986376, 1578280410414204

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026615, A026616, A026617, A026618, A026619, A026621, A026622.
Cf. A026623, A026624, A026625, A026956, A026957, A026958, A026959.
Cf. A026960.

[n eq 2 select 1 else (7*n^2-11*n+6)*Catalan(n)/(4*(2*n-1)): n in [2..40]]; // G. C. Greubel, Jun 13 2024

Table[(7*n^2-11*n+6)*Binomial[2*n,n]/(4*(n+1)*(2*n-1))-Boole[n==2], {n,2,40}] (* G. C. Greubel, Jun 13 2024 *)

[(7*n^2-11*n+6)*binomial(2*n,n)/(4*(n+1)*(2*n-1)) -int(n==2) for n in range(2,41)] # G. C. Greubel, Jun 13 2024

From G. C. Greubel, Jun 13 2024: (Start)
a(n) = (7*n^2 - 11*n + 6)*binomial(2*n, n)/(4*(n+1)*(2*n-1)) - [n=2].
G.f.: ( (2 - 7*x + 3*x^2) - (2 - 3*x + x^2 + 2*x^3)*sqrt(1-4*x) )/(2*x*sqrt(1-4*x)).
E.g.f.: (1/2)*exp(2*x)*( 3*(-1 + x)*BesselI(0, 2*x) + (4 - 3*x)*BesselI(1, 2*x) ) + (1/2)*(3 - x - x^2). (End)

A026621 a(n) = A026615(n, floor(n/2)).

Original entry on oeis.org

1, 1, 3, 5, 10, 17, 34, 60, 120, 217, 434, 798, 1596, 2970, 5940, 11154, 22308, 42185, 84370, 160446, 320892, 613054, 1226108, 2351440, 4702880, 9048522, 18097044, 34916300, 69832600, 135059220, 270118440, 523521630, 1047043260, 2033066025, 4066132050, 7908332190

Offset: 0

Clark Kimberling

nonn

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

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026622.
Cf. A026623, A026624, A026625, A026956, A026957, A026958, A026959.
Cf. A026960.

I:=[1,3]; [1] cat [n le 2 select I[n] else 2*((49*n^2-287*n+360 )*Self(n-1) + 2*(n-3)*(7*n-8)*(7*n-17)*Self(n-2) )/((n+1)*(7*n-24)*(7*n-15)) : n in [1..40]]; // G. C. Greubel, Jun 13 2024

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, 2*n-1, T[n -1,k-1] +T[n-1,k]]]; (* T = A026615 *)
Table[T[n, Floor[n/2]], {n,0,40}] (* G. C. Greubel, Jun 13 2024 *)

@CachedFunction
def T(n, k): # T = A026615
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return 2*n-1
    else: return T(n-1, k-1) + T(n-1, k)
def A026621(n): return T(n, int(n//2))
[A026621(n) for n in range(41)] # G. C. Greubel, Jun 13 2024

a(n) = 2*( (49*n^2 - 287*n + 360)*a(n-1) + 2*(n-3)*(7*n-8)*(7*n-17)*a(n-2) )/((n+1)*(7*n-24)*(7*n-15)) for n > 2. - G. C. Greubel, Jun 13 2024

A026623 a(n) = Sum_{k=0..floor(n/2)} A026615(n, k).

Original entry on oeis.org

1, 1, 4, 6, 18, 27, 72, 111, 283, 447, 1112, 1791, 4381, 7167, 17305, 28671, 68497, 114687, 271560, 458751, 1077949, 1835007, 4283069, 7340031, 17031503, 29360127, 67768777, 117440511, 269797323, 469762047, 1074583315, 1879048191

Offset: 0

Clark Kimberling

nonn

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

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026624, A026625, A026956, A026957, A026958, A026959.
Cf. A026960.

I:=[1,4,6]; [1] cat [n le 3 select I[n] else ( 2*(7*n-24)*(7*n-29)*(n-1)*Self(n-1) + 4*(7*n-8)*(7*n-31)*(n-3)*Self(n-2) - 8*(7*n-15)*(7*n-24)*(n-4)*Self(n-3) - (147*n^3 - 1603*n^2 + 5896*n - 7152))/(n*(7*n-31)*(7*n-22)): n in [1..41]]; // G. C. Greubel, Jun 15 2024

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, 2*n-1, T[n -1, k-1] + T[n-1,k]]]; (* T = A026615 *)
A026623[n_]:= Sum[T[n, k], {k, 0, Floor[n/2]}];
Table[A026623[n], {n,0,40}] (* G. C. Greubel, Jun 15 2024 *)

@CachedFunction
def T(n, k): # T = A026615
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return 2*n-1
    else: return T(n-1, k-1) + T(n-1, k)
def A026623(n): return sum(T(n,k) for k in range((n//2)+1))
[A026623(n) for n in range(41)] # G. C. Greubel, Jun 15 2024

a(n) = ( 2*(7*n-24)*(7*n-29)*(n-1)*a(n-1) + 4*(7*n-8)*(7*n-31)*(n-3)*a(n-2) - 8*(7*n-15)*(7*n-24)*(n-4)*a(n-3) - (147*n^3 - 1603*n^2 + 5896*n - 7152))/(n*(7*n-31)*(7*n-22)), for n > 3, with a(0) = a(1) = 1, a(2) = 4, and a(3) = 6. - G. C. Greubel, Jun 15 2024

A026624 a(n) = Sum_{j=0..n} Sum_{k=0..j} A026615(j, k).

Original entry on oeis.org

1, 3, 8, 20, 46, 100, 210, 432, 878, 1772, 3562, 7144, 14310, 28644, 57314, 114656, 229342, 458716, 917466, 1834968, 3669974, 7339988, 14680018, 29360080, 58720206, 117440460, 234880970, 469761992, 939524038, 1879048132

Offset: 0

Clark Kimberling

nonn

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

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026625, A026956, A026957, A026958, A026959.
Cf. A026960.

[1] cat [n le 1 select 3 else 2*Self(n-1) + 2*(n-1): n in [1..41]]; // G. C. Greubel, Jun 15 2024

Table[(7*2^n -4*(n+1) -Boole[n==0])/2, {n,0,40}] (* G. C. Greubel, Jun 15 2024 *)

[(7*2^n -4*(n+1) -int(n==0))/2 for n in range(41)] # G. C. Greubel, Jun 15 2024

G.f.: (1-x+x^2+x^3)/((1-x)^2*(1-2*x)) (Cf. A026622). - Ralf Stephan, Feb 05 2004
From G. C. Greubel, Jun 15 2024: (Start)
a(n) = 2*a(n-1) + 2*(n-1), with a(0) = 1, a(1) = 3.
a(n) = 7*2^(n-1) - 2*(n+1) - (1/2)*[n=0].
E.g.f.: (1/2)*( 7*exp(2*x) - 4*(x+1)*exp(x) - 1). (End)

A026625 a(n) = Sum_{k=0..floor(n/2)} A026615(n-k,k).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 21, 36, 58, 96, 155, 253, 409, 664, 1074, 1740, 2815, 4557, 7373, 11932, 19306, 31240, 50547, 81789, 132337, 214128, 346466, 560596, 907063, 1467661, 2374725, 3842388, 6217114, 10059504, 16276619, 26336125, 42612745, 68948872, 111561618

Offset: 0

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A000032, A000045, A026615, A026616, A026617, A026618, A026619.
Cf. A026620, A026621, A026622, A026623, A026624, A026956, A026957.
Cf. A026958, A026959, A026960.

[n eq 1 select 1 else 3*Fibonacci(n+1) - 2*Fibonacci(n) - (3+(-1)^n)/2: n in [0..40]]; // G. C. Greubel, Jun 16 2024

Join[{1,1},Table[Fibonacci[n-1]+LucasL[n]-(3+(-1)^n)/2,{n,2,40}]] (* or *) Join[{1,1},LinearRecurrence[{1,2,-1,-1},{2,4,7,13},40]] (* Harvey P. Dale, Sep 27 2011 *)

Vec((1 - x^2 + x^3 + x^4 + x^5) / ((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

[3*fibonacci(n+1) -2*fibonacci(n) -(3+(-1)^n)//2 + int(n==1) for n in range(41)] # G. C. Greubel, Jun 16 2024

For n>1, a(n) = Fibonacci(n-1) + Lucas(n) - (3 + (-1)^n)/2. - Ralf Stephan, May 13 2004
From Colin Barker, Jul 12 2017: (Start)
G.f.: (1 - x^2 + x^3 + x^4 + x^5) / ((1 - x)*(1 + x)*(1 - x - x^2)).
a(n) = 2^(-1-n)*(-5*((-2)^n + 3*2^n) - (-15+sqrt(5))*(1+sqrt(5))^n + (1-sqrt(5))^n*(15+sqrt(5))) / 5 for n>1.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>5.
(End)

cp's OEIS Frontend

A026622 a(n) = Sum_{k=0..n} A026615(n, k).

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A026615 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = 2*n-1 for n >= 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2 and n >= 4.

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A026617 a(n) = A026615(2*n, n-1).

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A026618 a(n) = A026618(2*n, n-2).

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A026619 a(n) = A026615(2*n-1, n-1).

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A026620 a(n) = A026615(2*n-1, n-2).

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A026621 a(n) = A026615(n, floor(n/2)).

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