A026736 -id:A026736 - cp's OEIS frontend

A026746 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026736.

1, 1, 2, 3, 5, 9, 14, 23, 42, 65, 107, 194, 301, 495, 890, 1385, 2275, 4058, 6333, 10391, 18404, 28795, 47199, 83079, 130278, 213357, 373512, 586869, 960381, 1673271, 2633652, 4306923, 7472326, 11779249, 19251575, 33275451, 52527026

Offset: 0

Clark Kimberling

nonn

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

Cf. A026736.

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([0..20], n-> Sum([0..Int(n/2)], k-> T(n-k,k) )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]]; Table[Sum[T[n-k, k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* G. C. Greubel, Jul 19 2019 *)

T(n,k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(30, n, n--; sum(k=0, n\2, T(n-k,k))) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n-k,k) for k in (0..floor(n/2))) for n in (0..40)] # G. C. Greubel, Jul 19 2019

a(n) ~ n * phi^(n-2) / 15, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019

A026850 a(n) = T(2n,n+1), T given by A026736.

Original entry on oeis.org

1, 4, 15, 57, 221, 872, 3489, 14113, 57575, 236457, 976271, 4047871, 16840879, 70259892, 293790127, 1230783085, 5164196117, 21696512073, 91254256589, 384165925259, 1618551762085, 6823801074549, 28785680471185, 121490461772347

Offset: 1

Clark Kimberling

nonn

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

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)*(1-Sqrt(1-4*x))^3/(8*x^2*(Sqrt(1-4*x) -x)) )); // G. C. Greubel, Jul 17 2019

CoefficientList[ Series[(1-Sqrt[1-4x])^3/(8x^3(1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* David Callan, Jan 16 2016 *)

my(x='x+O('x^30)); Vec(sqrt(1-4*x)*(1-sqrt(1-4*x))^3/(8*x^2*(sqrt(1-4*x) -x)) ) \\ G. C. Greubel, Jul 17 2019

a=(sqrt(1-4*x)*(1-sqrt(1-4*x))^3/(8*x^2*(sqrt(1-4*x) -x))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 17 2019

G.f.: (x * C(x)^3)/(1 - x/sqrt(1 - 4 * x)) where C(x) is the g.f. for Catalan numbers A000108. - David Callan, Jan 16 2016

a(n) ~ (3 - sqrt(5))^3 * (2 + sqrt(5))^(n+1) / (8*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019

A026851 a(n) = T(2n,n+2), T given by A026736.

Original entry on oeis.org

1, 6, 28, 121, 508, 2109, 8723, 36065, 149277, 618961, 2571503, 10704390, 44641793, 186492242, 780275596, 3269135406, 13713525610, 57588530626, 242068874444, 1018378855512, 4287501276956, 18062827159136, 76141329903018

Offset: 2

Clark Kimberling

nonn

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

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)*(1-Sqrt(1-4*x))^5/(32*x^3*(Sqrt(1-4*x) -x)) )); // G. C. Greubel, Jul 17 2019

CoefficientList[Series[(1-Sqrt[1-4x])^5/(32 x^5 (1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* David Callan, Jan 16 2016 *)

my(x='x+O('x^30)); Vec(sqrt(1-4*x)*(1-sqrt(1-4*x))^5/(32*x^3*(sqrt(1-4*x) -x)) ) \\ G. C. Greubel, Jul 17 2019

a=(sqrt(1-4*x)*(1-sqrt(1-4*x))^5/(32*x^3*(sqrt(1-4*x)-x))).series(x, 30).coefficients(x, sparse=False); a[2:] # G. C. Greubel, Jul 17 2019

G.f.: (x * C(x)^5)/(1 - x/sqrt(1 - 4 * x)) where C(x) is the g.f. for Catalan numbers A000108. - David Callan, Jan 16 2016
a(n) ~ (3 - sqrt(5))^5 * (2 + sqrt(5))^(n+2) / (32*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019
D-finite with recurrence -(n+3)*(55*n-136)*a(n) +2*(331*n^2-427*n-1440)*a(n-1) +3*(-867*n^2+2755*n-252)*a(n-2) +2*(1555*n^2-8227*n+11304)*a(n-3) +12*(37*n-89)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Nov 22 2024

A026853 a(n) = T(2n,n+4), T given by A026736.

Original entry on oeis.org

1, 10, 66, 365, 1837, 8741, 40133, 179932, 793605, 3460106, 14961664, 64306917, 275180827, 1173714565, 4994096327, 21211537533, 89972566673, 381261067469, 1614446775255, 6832832045575, 28908094009481, 122272843951891, 517095189163181

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..1000

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( Sqrt(1-4*x)*(1-Sqrt(1-4*x))^9/(512*x^5*(Sqrt(1-4*x) -x)) )); // G. C. Greubel, Jul 17 2019

Drop[CoefficientList[Series[Sqrt[1-4x]*(1-Sqrt[1-4x])^9/(512*x^5*(Sqrt[1-4x]-x)), {x,0,40}], x], 4] (* G. C. Greubel, Jul 17 2019 *)

my(x='x+O('x^40)); Vec(sqrt(1-4*x)*(1-sqrt(1-4*x))^9/(512*x^5*(sqrt(1-4*x) -x)) ) \\ G. C. Greubel, Jul 17 2019

a=(sqrt(1-4*x)*(1-sqrt(1-4*x))^9/(512*x^5*(sqrt(1-4*x)-x))).series(x, 40).coefficients(x, sparse=False); a[4:] # G. C. Greubel, Jul 17 2019

G.f.: x^4*C(x)^9/(1 -x/sqrt(1-4*x)), where C(x) if the g.f. for Catalan numbers A000108. - G. C. Greubel, Jul 17 2019
a(n) ~ (3 - sqrt(5))^9 * (2 + sqrt(5))^(n+4) / (512*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019
D-finite with recurrence -(n+5)*(3013*n-9152)*a(n) +2*(15151*n^2+5919*n-112720)*a(n-1) +2*(-35029*n^2+9054*n-235442)*a(n-2) +6*(-19475*n^2+144598*n+188045)*a(n-3) +3*(131869*n^2-942353*n+1922276)*a(n-4) +2*(2*n-7)*(24721*n-92359)*a(n-5)=0. - R. J. Mathar, Nov 22 2024

A026854 a(n) = T(2n+1,n+1), T given by A026736.

Original entry on oeis.org

1, 3, 10, 36, 136, 530, 2109, 8515, 34739, 142817, 590537, 2452639, 10221505, 42714623, 178888442, 750500716, 3153137436, 13263180550, 55844218906, 235323138044, 992316962382, 4186870456952, 17674378119680, 74641954142026

Offset: 0

Clark Kimberling

nonn

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

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-Sqrt(1-4*x))^2/(4*x^2*(1-x/Sqrt(1-4*x))) )); // G. C. Greubel, Jul 21 2019

CoefficientList[Series[(1-Sqrt[1-4x])^2/(4x^2(1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* David Callan, Jan 16 2016 *)

my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^2/(4*x^2*(1-x/sqrt(1-4*x))) ) \\ G. C. Greubel, Jul 21 2019

((1-sqrt(1-4*x))^2/(4*x^2*(1-x/sqrt(1-4*x)))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 21 2019

G.f.: C(x)^2/(1 - x/sqrt(1-4*x)) where C(x) = g.f. for Catalan numbers A000108. - David Callan, Jan 16 2016

a(n) ~ (3 - sqrt(5))^2 * (2 + sqrt(5))^(n+1) / (4*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019

A026855 a(n) = T(2n+1,n+2), T given by A026736.

Original entry on oeis.org

1, 5, 21, 85, 342, 1380, 5598, 22836, 93640, 385734, 1595232, 6619374, 27545269, 114901685, 480282369, 2011058681, 8433331523, 35410037683, 148842787215, 626234799703, 2636930617597, 11111302351505, 46848507630321, 197631791675365

Offset: 1

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 1..1580

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)*(1-Sqrt(1-4*x))^4/(16*x^3*(Sqrt(1-4*x) -x)) )); // G. C. Greubel, Jul 17 2019

gf := ((-2*x^3+12*x^2-7*x+1)*sqrt(1-4*x)+16*x^3-24*x^2+9*x-1)/(2*(x^2+4*x-1)*x^3):
S:= series(gf,x,40):
seq(coeff(S,x,j),j=1..30); # Robert Israel, Jan 17 2016

CoefficientList[Series[(1-Sqrt[1-4x])^4/(16*x^4*(1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* David Callan, Jan 16 2016 *)

my(x='x+O('x^30)); Vec( sqrt(1-4*x)*(1-sqrt(1-4*x))^4/(16*x^3*(sqrt(1-4*x) -x)) ) \\ G. C. Greubel, Jul 17 2019

a=(sqrt(1-4*x)*(1-sqrt(1-4*x))^4/(16*x^3*(sqrt(1-4*x) -x))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 17 2019

G.f.: (x*C(x)^4)/(1 - x/sqrt(1 - 4*x)) where C(x) is the g.f. for Catalan numbers A000108. - David Callan, Jan 16 2016
a(n) ~ (3 - sqrt(5))^4 * (2 + sqrt(5))^(n+2) / (16*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019
D-finite with recurrence -2*(n+3)*(26*n-53)*a(n) +(627*n^2-491*n-2440)*a(n-1) +2*(-1234*n^2+3198*n+337)*a(n-2) +(2957*n^2-13637*n+15888)*a(n-3) +2*(211*n-368)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Nov 22 2024

A026856 a(n) = T(2n+1,n+3), T given by A026736.

Original entry on oeis.org

1, 7, 36, 166, 729, 3125, 13229, 55637, 233227, 976271, 4085016, 17096524, 71590557, 299993227, 1258076725, 5280194087, 22178492943, 93226087229, 392144055809, 1650570659359, 6951524807631, 29292822272697, 123496979334851

Offset: 2

Clark Kimberling

nonn

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

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-Sqrt(1-4*x))^6/(64*x^6*(1-x/Sqrt(1-4*x))) )); // G. C. Greubel, Jul 21 2019

CoefficientList[Series[(1-Sqrt[1-4x])^6/(64*x^6*(1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* David Callan, Jan 16 2016 *)

my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^6/(64*x^6*(1-x/sqrt(1-4*x))) ) \\ G. C. Greubel, Jul 21 2019

((1-sqrt(1-4*x))^6/(64*x^6*(1-x/sqrt(1-4*x)))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 21 2019

G.f.: (x^2 * C(x)^6)/(1 - x/sqrt(1-4*x)) where C(x) = g.f. for Catalan numbers A000108. - David Callan, Jan 16 2016

a(n) ~ (3 - sqrt(5))^6 * (2 + sqrt(5))^(n+3) / (64*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019

A026857 a(n) = T(2n+1,n+4), T given by A026736.

Original entry on oeis.org

1, 9, 55, 287, 1381, 6343, 28313, 124083, 537242, 2307118, 9852240, 41910428, 177807902, 752981956, 3184773246, 13459063660, 56849094136, 240047748038, 1013452871316, 4278470305930, 18062827159136, 76263743441314, 322033566728056

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-4*x))^8/(2^8*x^5*(1-x/Sqrt(1-4*x))) )); // G. C. Greubel, Jul 19 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4x])^8/(2^8*x^5*(1-x/Sqrt[1-4x])), {x, 0, 40}], x], 3] (* G. C. Greubel, Jul 19 2019 *)

my(x='x+O('x^40)); Vec((1-sqrt(1-4*x))^8/(2^8*x^5*(1-x/sqrt(1-4*x)))) \\ G. C. Greubel, Jul 19 2019

a=((1-sqrt(1-4*x))^8/(2^8*x^5*(1-x/sqrt(1-4*x)))).series(x, 45).coefficients(x, sparse=False); a[3:40] # G. C. Greubel, Jul 19 2019

G.f.: x^3*C(x)^8/(1 - x/sqrt(1-4*x)). - G. C. Greubel, Jul 19 2019

a(n) ~ phi^(3*n-4) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019

A027215 Self-convolution of row n of array T given by A026736.

Original entry on oeis.org

1, 2, 6, 20, 78, 282, 1187, 4428, 19175, 72820, 319493, 1227712, 5424359, 21018514, 93252862, 363563668, 1617342486, 6334904252, 28232695584, 110982722888, 495257577162

Offset: 0

Clark Kimberling

nonn

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

Cf. A026736.

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([0..20], n-> Sum([0..n], k-> T(n, k)*T(n,n-k) )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]];
Table[Sum[T[n, k]*T[n, n-k], {k,0,n}], {n,0,40}] (* G. C. Greubel, Jul 19 2019 *)

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(21, n, n--; sum(k=0, n, T(n, k)*T(n,n-k)) ) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    if (k==n-1): return n
    elif (mod(n, 2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n,k)*T(n,n-k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, Jul 19 2019

A027216 a(n) = Sum_{k=0..n-1} T(n,k)*T(n,k+1), T given by A026736.

Original entry on oeis.org

1, 4, 15, 63, 237, 1034, 3945, 17577, 67640, 304902, 1179415, 5352038, 20771331, 94628132, 368083879, 1680820301, 6548692260, 29946087674, 116816782997, 534628747310

Offset: 1

Clark Kimberling

nonn

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

Cf. A026736.

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([1..20], n-> Sum([0..n-1], k-> T(n, k)*T(n,k+1) )); # G. C. Greubel, Jul 19 2019

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/2, T[n-1,k-1] + T[n-2,k-1] + T[n-1,k], T[n-1,k-1] + T[n-1,k]]]; Table[Sum[T[n,k]*T[n,k+1], {k, 0, n-1}], {n, 1, 30}] (* G. C. Greubel, Jul 19 2019 *)

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(20, n, sum(k=0, n-1, T(n, k)*T(n,k+1)) ) \\ G. C. Greubel, Jul 19 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n,k)*T(n,k+1) for k in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jul 19 2019

a(n) ~ (1/2 - (-1)^n/10) * phi^(3*n - 5/2 + (-1)^n/2), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019

cp's OEIS Frontend

A026746 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026736.

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A026850 a(n) = T(2n,n+1), T given by A026736.

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A026851 a(n) = T(2n,n+2), T given by A026736.

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A026853 a(n) = T(2n,n+4), T given by A026736.

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A026854 a(n) = T(2n+1,n+1), T given by A026736.

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Magma

Mathematica

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A026855 a(n) = T(2n+1,n+2), T given by A026736.

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A026856 a(n) = T(2n+1,n+3), T given by A026736.

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