A026736 -id:A026736 - cp's OEIS frontend

A026849 a(n) = T(2n,n-3), T given by A026736.

1, 9, 56, 300, 1487, 7041, 32381, 146017, 649395, 2859231, 12494914, 54291912, 234860677, 1012433965, 4352210327, 18666918033, 79916230409, 341615895659, 1458457275715, 6220016154525, 26503542364381, 112847001503099, 480173686483581

Offset: 3

Clark Kimberling

nonn

Is this the same as A026846 and A026842? - R. J. Mathar, Oct 23 2008

Column k=8 of triangle A236830. - Philippe Deléham, Feb 02 2014

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

Cf. A236830.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-Sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1-Sqrt(1-4*x))^3 )) )); // G. C. Greubel, Jul 17 2019

CoefficientList[Series[(1-Sqrt[1-4*x])^8/(32*x^3*(8*x^2 -(1-Sqrt[1-4*x])^3 )), {x,0,30}], x] (* G. C. Greubel, Jul 17 2019 *)

my(x='x+O('x^30)); Vec((1-sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1-sqrt(1-4*x))^3 ))) \\ G. C. Greubel, Jul 17 2019

a=((1-sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1-sqrt(1-4*x))^3 ))).series(x, 30).coefficients(x, sparse=False); a[3:] # G. C. Greubel, Jul 17 2019

a(n) = A026842(n) = A026846(n). - Philippe Deléham, Feb 02 2014

G.f.: (x^3*C(x)^8)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

A026737 a(n) = T(2*n,n), T given by A026736.

Original entry on oeis.org

1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550, 456710589, 1922354351, 8098984433, 34147706833, 144068881455, 608151037123, 2568318694867, 10850577045131, 45856273670841

Offset: 0

Clark Kimberling

nonn

Is this the same sequence as A111279? - Andrew S. Plewe, May 09 2007

Yes, see the Callan reference "A bijection...". - Joerg Arndt, Feb 29 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Callan, A bijection for two sequences in OEIS, arXiv:1602.08347 [math.CO], 2016.

Cf. A026736, A111279, A059714.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-5*x+4*x^2 -(1-5*x)*Sqrt(1-4*x))/(2*x*(1-4*x-x^2)) )); // G. C. Greubel, Jul 16 2019

T[, 0]=T[n, n_]=1; T[n_, k_]:= T[n, k]= 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]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 22 2018 *)
CoefficientList[Series[(1-5x+4x^2 -(1-5x)*Sqrt[1-4x])/(2*x*(1-4x-x^2)), {x, 0, 30}], x] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec((1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2))) \\ G. C. Greubel, Jul 16 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)
[T(2*n, n) for n in (0..30)] # G. C. Greubel, Jul 16 2019

G.f.: (1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2)). - G. C. Greubel, Jul 16 2019
a(n) ~ (47 - 21*sqrt(5)) * (2 + sqrt(5))^(n+2) / (2*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019
G.f. G satisfies 0 = G^2*(x^3 + 4*x^2 - x) + G*(4*x^2 - 5*x + 1) + 4*x - 1. - F. Chapoton, Oct 16 2021

A026848 a(n) = T(2n,n-4), T given by A026736.

Original entry on oeis.org

1, 11, 79, 471, 2535, 12809, 62067, 292085, 1345718, 6102780, 27343148, 121359692, 534632836, 2341151646, 10201950700, 44278673806, 191540714294, 826265471868, 3555992623850, 15273547250820, 65491352071266, 280412963707416

Offset: 4

Clark Kimberling

nonn

Is this the same as A026841? - R. J. Mathar, Oct 23 2008

Column k=10 of triangle A236830. - Philippe Deléham, Feb 02 2014

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

Cf. A236830.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-4*x))^10/(128*x^4*(8*x^2 -(1-Sqrt(1-4*x))^3 )) )); // G. C. Greubel, Jul 17 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^10/(128*x^4*(8*x^2 -(1 - Sqrt[1-4*x])^3 )), {x,0,40}], x], 4] (* G. C. Greubel, Jul 17 2019 *)

my(x='x+O('x^40)); Vec((1-sqrt(1-4*x))^10/(128*x^4*(8*x^2 -(1 - sqrt(1-4*x))^3 ))) \\ G. C. Greubel, Jul 17 2019

a=((1-sqrt(1-4*x))^10/(128*x^4*(8*x^2 -(1-sqrt(1-4*x))^3 ))).series(x, 45).coefficients(x, sparse=False); a[4:40] # G. C. Greubel, Jul 17 2019

a(n) = A026841(n). - Philippe Deléham, Feb 02 2014

G.f.: (x^4*C(x)^10)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

A026672 a(n) = T(2n,n-1), T given by A026670. Also T(2n,n-1)=T(2n+1,n+2), T given by A026725; and T(2n,n-1), T given by A026736.

Original entry on oeis.org

1, 5, 22, 94, 398, 1680, 7085, 29877, 126021, 531751, 2244627, 9478605, 40040183, 169193597, 715143046, 3023492646, 12785541850, 54076955716, 228759017624, 967850695362, 4095387893312, 17331318506030

Offset: 2

Clark Kimberling

nonn

Column k=4 of triangle A236830. - Philippe Deléham, Feb 02 2014

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

Cf. A000108, A026670, A026725, A026736, A236830.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(  (1-Sqrt(1-4*x))^4/(2*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 16 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^4/(2*(8*x^2 -(1-Sqrt[1-4*x] )^3)), {x,0,30}], x], 2] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^4/(2*(8*x^2 -(1-sqrt(1-4*x))^3))) \\ G. C. Greubel, Jul 16 2019

a=((1-sqrt(1-4*x))^4/(2*(8*x^2 -(1-sqrt(1-4*x))^3))).series(x, 30).coefficients(x, sparse=False); a[2:] # G. C. Greubel, Jul 16 2019

G.f.: (x*C(x)^4)/(1-x*C(x)^3), where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

Conjecture: -(n+1)*(n-6)*a(n) +2*(4*n^2-23*n+3)*a(n-1) +3*(-5*n^2+33*n-42)*a(n-2) -2*(2*n-3)*(n-5)*a(n-3)=0. - R. J. Mathar, Aug 08 2015

A026675 a(n) = T(2n-1,n-2), T given by A026670. Also T(2n-1,n-2) = T(2n,n+2), T given by A026725 and T(2n,n-2), T given by A026736.

Original entry on oeis.org

1, 6, 29, 131, 575, 2488, 10681, 45641, 194467, 827045, 3512983, 14909339, 63239487, 268127302, 1136495965, 4816202207, 20406887583, 86457399359, 366263778659, 1551535465465, 6572224024539, 27838835937511, 117918419518219

Offset: 2

Clark Kimberling

nonn

Column k=5 of triangle A236830. - Philippe Deléham, Feb 02 2014

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

Cf. A000108, A026670, A026725, A026736, A236830.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(  (1-Sqrt(1-4*x))^5/(4*x*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 16 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^5/(4*x*(8*x^2 -(1-Sqrt[1 - 4*x])^3)), {x,0,30}], x], 2] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^5/(4*x*(8*x^2 -(1-sqrt(1-4*x))^3))) \\ G. C. Greubel, Jul 16 2019

a=((1-sqrt(1-4*x))^5/(4*x*(8*x^2 -(1-sqrt(1-4*x))^3))).series(x, 30).coefficients(x, sparse=False); a[2:] # G. C. Greubel, Jul 16 2019

G.f.: (x^2*C(x)^5)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

A026743 a(n) = Sum_{j=0..n} T(n,j), T given by A026736.

Original entry on oeis.org

1, 2, 4, 8, 17, 34, 73, 146, 314, 628, 1350, 2700, 5798, 11596, 24872, 49744, 106573, 213146, 456169, 912338, 1950697, 3901394, 8334539, 16669078, 35582783, 71165566, 151809737, 303619474, 647279131, 1294558262, 2758310121

Offset: 0

Clark Kimberling

nonn

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

Cf. A026736.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( ((1 -3*x^2)*Sqrt((1+2*x)/(1-2*x)) +(1+2*x)*(1+x^2))/(2*(1-4*x^2-x^4)) )); // G. C. Greubel, Jul 16 2019

CoefficientList[Normal[Series[((1-3x^2)Sqrt[(1+2x)/(1-2x)] +(1 + 2x)(1+ x^2))/(2(1-4x^2-x^4)), {x,0,40}]], x] (* David Callan, Jan 17 2016 *)

my(x='x+O('x^40)); Vec(((1-3*x^2)*sqrt((1+2*x)/(1-2*x)) +(1+2*x)*(1+x^2))/(2*(1-4*x^2-x^4))) \\ G. C. Greubel, Jul 16 2019

(((1-3*x^2)*sqrt((1+2*x)/(1-2*x)) + (1+2*x)*(1+x^2))/(2*(1-4*x^2 - x^4))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 16 2019

G.f.: ((1-3*x^2)*sqrt((1+2*x)/(1-2*x)) + (1+2*x)*(1+x^2))/(2*(1 -4*x^2 - x^4)). - David Callan, Jan 17 2016
Conjecture D-finite with recurrence n*a(n) -2*a(n-1) +(-11*n+20)*a(n-2) +14*a(n-3) +(39*n-152)*a(n-4) -22*a(n-5) +(-41*n+268)*a(n-6) -6*a(n-7) +12*(-n+6)*a(n-8)=0. - R. J. Mathar, Jan 13 2023
a(n) ~ ((1 + (-1)^n)*phi^(3/2) + 2*(1 - (-1)^n)) * phi^((3*n + 1)/2) / (2*sqrt(5)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 08 2023

A026852 a(n) = T(2n,n+3), T given by A026736.

Original entry on oeis.org

1, 8, 45, 221, 1016, 4506, 19572, 83950, 357310, 1513513, 6392134, 26948764, 113500985, 477801129, 2011058681, 8464967333, 35637556603, 150075181365, 632191803847, 2664023530675, 11229995113561, 47355649431833, 199760722776165

Offset: 3

Clark Kimberling

nonn

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

Cf. A000108, A026736.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)*(1-Sqrt(1-4*x))^9/(64*x^4*(8*x^2 -(1-Sqrt(1-4*x))^3 )) )); // G. C. Greubel, Jul 17 2019

Drop[CoefficientList[Series[Sqrt[1-4*x]*(1-Sqrt[1-4*x])^9/(64*x^4*(8*x^2 -(1-Sqrt[1-4*x])^3)), {x, 0, 40}], x], 3] (* G. C. Greubel, Jul 17 2019 *)

my(x='x+O('x^40)); Vec(sqrt(1-4*x)*(1-sqrt(1-4*x))^9/(64*x^4*(8*x^2 -(1 - sqrt(1-4*x))^3 ))) \\ G. C. Greubel, Jul 17 2019

a=(sqrt(1-4*x)*(1-sqrt(1-4*x))^9/(64*x^4*(8*x^2 -(1-sqrt(1-4*x))^3 ))).series(x, 45).coefficients(x, sparse=False); a[3:40] # G. C. Greubel, Jul 17 2019

G.f.: x^3*C(x)^7/(1 - x/Sqrt(1-4*x)) = x^3*(1-2*x*C(x))*C(x)^9/(1-x*C(x)^3), where C(x) is the g.f. of A000108. - G. C. Greubel, Jul 17 2019

a(n) ~ (2 + sqrt(5))^(n+3) * (3 - sqrt(5))^7 / (128*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019

A026742 a(n) = T(n, floor(n/2)), T given by A026736.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 21, 43, 79, 173, 309, 707, 1237, 2917, 5026, 12111, 20626, 50503, 85242, 211263, 354080, 885831, 1476368, 3720995, 6173634, 15652239, 25873744, 65913927, 108628550, 277822147, 456710589, 1171853635, 1922354351

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;
Flat(List([0..20], n-> T(n,Int(n/2)) )); # 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[T[n, Floor[n/2]], {n,0,40}] (* G. C. Greubel, Jul 19 2019 *)

T(n,k) = if(k==n || k==0, 1, 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, n--; T(n, n\2)) \\ 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)
[T(n, floor(n/2)) for n in (0..40)] # G. C. Greubel, Jul 19 2019

a(n) ~ phi^(3*n/2 - (7 + (-1)^n)/4) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019

A026744 a(n) = Sum_{j=0..floor(n/2)} T(n,j), T given by A026736.

Original entry on oeis.org

1, 1, 3, 4, 12, 18, 51, 81, 220, 361, 952, 1595, 4118, 6999, 17787, 30548, 76696, 132766, 330148, 575054, 1418946, 2483812, 6089912, 10703456, 26104178, 46034722, 111769554, 197665364, 478085534, 847542518, 2043167075

Offset: 0

Clark Kimberling

nonn

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

Cf. A026736.

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, j], {j, 0, Floor[n/2]}], {n, 0, 35}] (* G. C. Greubel, Jul 22 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, j) for j in (0..floor(n/2))) for n in (0..35)] # G. C. Greubel, Jul 22 2019

A026745 a(n) = Sum_{j=0..n} Sum_{i=0..n} T(j,i), T given by A026736.

Original entry on oeis.org

1, 3, 7, 15, 32, 66, 139, 285, 599, 1227, 2577, 5277, 11075, 22671, 47543, 97287, 203860, 417006, 873175, 1785513, 3736210, 7637604, 15972143, 32641221, 68224004, 139389570, 291199307, 594818781, 1242097912, 2536656174

Offset: 0

Clark Kimberling

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1000 from G. C. Greubel)

Cf. A026736.

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]]];
b[n_]:= Sum[T[n, j], {j,0,n}]; Table[Sum[b[j], {j,0,n}], {n,0,35}] (* G. C. Greubel, Jul 22 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)
def b(n): return sum(T(n, j) for j in (0..n))
[sum(b(j) for j in (0..n)) for n in (0..35)] # G. C. Greubel, Jul 22 2019

a(n) ~ c * phi^(3*n/2), where c = 1/2 + 3*phi^2 / (2*sqrt(5)) if n is even, c = 3*phi^(5/2) / (2*sqrt(5)) if n is odd and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 22 2019

cp's OEIS Frontend

A026849 a(n) = T(2n,n-3), T given by A026736.

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A026737 a(n) = T(2*n,n), T given by A026736.

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A026848 a(n) = T(2n,n-4), T given by A026736.

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A026672 a(n) = T(2n,n-1), T given by A026670. Also T(2n,n-1)=T(2n+1,n+2), T given by A026725; and T(2n,n-1), T given by A026736.

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A026675 a(n) = T(2n-1,n-2), T given by A026670. Also T(2n-1,n-2) = T(2n,n+2), T given by A026725 and T(2n,n-2), T given by A026736.

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A026743 a(n) = Sum_{j=0..n} T(n,j), T given by A026736.

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A026852 a(n) = T(2n,n+3), T given by A026736.

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