cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

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

Views

Author

Clark Kimberling

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

A026673 a(n) = T(2n,n-2), T given by A026670.

Original entry on oeis.org

1, 7, 37, 177, 808, 3596, 15764, 68446, 295294, 1268356, 5430734, 23199304, 98933705, 421352919, 1792709561, 7621345733, 32380443643, 137504761035, 583684770103, 2476836131227, 10507517431481, 44566369523517, 188988331406117
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

Also a(n) = T(2n,n-2) = T(2n+1,n+2), T given by A026725.
Also a(n) = T(2n,n-2), T given by A026736.
Column k=6 of triangle A236830. - Philippe Deléham, Feb 02 2014

Links

Crossrefs

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

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(  (1-Sqrt(1-4*x))^6/(8*x^2*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 16 2019
  • Mathematica
    Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^6/(8*x^2*(8*x^2-(1-Sqrt[1 - 4*x])^3)), {x,0,30}], x], 2] (* G. C. Greubel, Jul 16 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^6/(8*x^2*(8*x^2 -(1-sqrt(1-4*x))^3))) \\ G. C. Greubel, Jul 16 2019
  • Sage
    a=((1-sqrt(1-4*x))^6/(8*x^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

Formula

G.f.: (x^2*C(x)^6)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014
-(n+2)*(3*n-7)*a(n) +2*(12*n^2-19*n-16)*a(n-1) +5*(-9*n^2+27*n-22)*a(n-2) -2*(3*n-4)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Oct 26 2019

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

Views

Author

Clark Kimberling

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

A026676 a(n) = T(n, floor(n/2)), T given by A026670.

Original entry on oeis.org

1, 1, 3, 4, 11, 16, 43, 65, 173, 267, 707, 1105, 2917, 4597, 12111, 19196, 50503, 80380, 211263, 337284, 885831, 1417582, 3720995, 5965622, 15652239, 25130844, 65913927, 105954110, 277822147, 447015744, 1171853635, 1886996681
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Also a(n) = T(n,m) + T(n,m+1) + ... + T(n,n), m=[ (n+1)/2 ], T given by A026736.

Links

Crossrefs

Cf. A026670, A026736.

Programs

  • GAP
    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([Int((n+1)/2)..n], k-> T(n, k) )); # G. C. Greubel, Jul 19 2019
  • Mathematica
    T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[k==n-1, n, 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, Floor[(n+1)/2], n}], {n, 0, 40}] (* G. C. Greubel, Jul 19 2019 *)
  • PARI
    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, n--; sum(k=(n+1)\2, n, T(n, k)) ) \\ G. C. Greubel, Jul 19 2019
  • Sage
    @CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (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) for k in (floor((n+1)/2)..n)) for n in (0..40)] # G. C. Greubel, Jul 19 2019

A026677 T(n,0) + T(n,1) + ... + T(n,n), T given by A026670.

Original entry on oeis.org

1, 2, 5, 10, 23, 46, 103, 206, 455, 910, 1993, 3986, 8679, 17358, 37633, 75266, 162643, 325286, 701075, 1402150, 3015563, 6031126, 12948083, 25896166, 55513327, 111026654, 237705547, 475411094, 1016736115, 2033472230
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Partial sums of A026733.

Formula

G.f.: F(x^2) + 2xF(x^2), where xF(x) = g.f. of A054441. - Ralf Stephan, Feb 05 2004

A026678 a(n) = T(n,0) + T(n,1) + ... + T(n,[ n/2 ]), T given by A026670.

Original entry on oeis.org

1, 1, 4, 5, 17, 23, 73, 103, 314, 455, 1350, 1993, 5798, 8679, 24872, 37633, 106573, 162643, 456169, 701075, 1950697, 3015563, 8334539, 12948083, 35582783, 55513327, 151809737, 237705547, 647279131, 1016736115, 2758310121
Offset: 0

Views

Author

Clark Kimberling

Keywords

A026679 Sum{T(i,j)}, 0<=i<=n, 0<=j<=n, T given by A026670.

Original entry on oeis.org

1, 3, 8, 18, 41, 87, 190, 396, 851, 1761, 3754, 7740, 16419, 33777, 71410, 146676, 309319, 634605, 1335680, 2737830, 5753393, 11784519, 24732602, 50628768, 106142095, 217168749, 454874296, 930285390, 1947021505
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Partial sums of A026677.

A026680 Sum{T(n-k,k)}, 0<=k<=[ n/2 ], T given by A026670.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 19, 29, 48, 88, 136, 224, 403, 627, 1030, 1830, 2860, 4690, 8257, 12947, 21204, 37068, 58272, 95340, 165723, 261063, 426786, 738352, 1165138, 1903490, 3279891, 5183381, 8463272, 14532484, 22995756, 37528240
Offset: 0

Views

Author

Clark Kimberling

Keywords

A026981 Self-convolution of array T given by A026670.

Original entry on oeis.org

1, 2, 11, 34, 173, 586, 2917, 10262, 50503, 181334, 885831, 3221146, 15652239, 57405038, 277822147, 1025177314, 4945846997, 18333973274, 88224662549, 328195843910, 1576001732485, 5879010027618, 28181895551161, 105362519875518, 504329070986033, 1888952213601906
Offset: 0

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026982 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026670.

Original entry on oeis.org

1, 6, 24, 120, 460, 2214, 8475, 40164, 154235, 724742, 2792579, 13050280, 50432693, 234766510, 909442512, 4221139640, 16384219216, 75873259406, 294977735364, 1363503281324, 5308113306398, 24499364386598, 95483014454706, 440147035814920, 1717035728143720, 7906666535747934
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 20 2019
Showing 1-10 of 16 results. Next

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