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 1-6 of 6 results.

A127357 Expansion of 1/(1 - 2*x + 9*x^2).

Original entry on oeis.org

1, 2, -5, -28, -11, 230, 559, -952, -6935, -5302, 51811, 151340, -163619, -1689298, -1906025, 11391632, 39937489, -22649710, -404736821, -605626252, 2431378885, 10313394038, -1255621889
Offset: 0

Views

Author

Paul Barry, Jan 11 2007

Keywords

Comments

Hankel transform of A100193. A member of the family of sequences with g.f. 1/(1-2*x+r^2*x^2) which are the Hankel transforms of the sequences given by Sum_{k=0..n} binomial(2*n,k)*r^(n-k).
From Peter Bala, Apr 01 2018: (Start)
With offset 1, this is the Lucas sequence U(n,2,9). The companion Lucas sequence V(n,2,9) is 2*A025172(n).
Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 2 o 2 o ... o 2 (n terms) = 2*A127357(n-1)/A025172(n). Cf. A088137 and A087455. (End)

Links

Crossrefs

Variant is A025170.
Cf. A025172, A087455, A088137, A100193, A127357.

Programs

  • GAP
    a:=[1,2];; for n in [3..25] do a[n]:=2*a[n-1]-9*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018
  • Magma
    m:=23; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-2*x+9*x^2))); // Bruno Berselli, Jul 01 2011
  • Magma
    [3^n*Evaluate(ChebyshevU(n+1),1/3): n in [0..50]]; // G. C. Greubel, Jan 02 2024
  • Maple
    c := 2*sqrt(2): g := exp(x)*(sin(c*x)+c*cos(c*x))/c: ser := series(g,x,32):
seq(n!*coeff(ser,x,n), n=0..22); # Peter Luschny, Oct 19 2016
  • Mathematica
    RootReduce@Table[3^n (Cos[n ArcTan[2 Sqrt[2]]] + Sin[n ArcTan[2 Sqrt[2]]] Sqrt[2]/4), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)
CoefficientList[Series[1/(1-2x+9x^2),{x,0,40}],x] (* or *)
LinearRecurrence[ {2,-9},{1,2},40] (* Harvey P. Dale, Mar 15 2022 *)
Table[3^n*ChebyshevU[n, 1/3], {n,0,40}] (* G. C. Greubel, Jan 02 2024 *)
  • Maxima
    makelist(coeff(taylor(1/(1-2*x+9*x^2), x, 0, n), x, n), n, 0, 22); /* Bruno Berselli, Jul 01 2011 */
  • PARI
    Vec(1/(1-2*x+9*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
  • Sage
    [lucas_number1(n,2,9) for n in range(1, 24)] # Zerinvary Lajos, Apr 23 2009
  • SageMath
    [3^n*chebyshev_U(n,1/3) for n in range(41)] # G. C. Greubel, Jan 02 2024

Formula

a(n) = Sum_{k=0..n} binomial(n-k,k)*2^(n-2*k)*(-9)^k.
a(n) = 2*a(n-1) - 9*a(n-2) for n >= 2. - Vincenzo Librandi, Mar 22 2011
a(n) = ((1-2*sqrt(2)*i)^n-(1+2*sqrt(2)*i)^n)*i/(4*sqrt(2)), where i=sqrt(-1). - Bruno Berselli, Jul 01 2011
From Vladimir Reshetnikov, Oct 15 2016: (Start)
a(n) = 3^n*(cos(n*theta) + sin(n*theta)*sqrt(2)/4), theta = arctan(2*sqrt(2)).
E.g.f.: exp(x)*(cos(2*sqrt(2)*x) + sin(2*sqrt(2)*x)*sqrt(2)/4). (End)
a(n) = 2^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
From G. C. Greubel, Jan 02 2024: (Start)
a(n) = (-1)^n * A025170(n).
a(n) = 3^n * ChebyshevU(n, 1/3). (End)

A293574 a(n) = Sum_{k=0..n} n^(n-k)*binomial(n+k-1,k).

Original entry on oeis.org

1, 2, 11, 82, 787, 9476, 139134, 2422218, 48824675, 1118286172, 28679699578, 814027423892, 25330145185646, 857375286365768, 31360145331198428, 1232586016712594010, 51805909208539809315, 2318588202311267591852, 110085368092924083334626, 5526615354023679440754396, 292501304641192746350100410
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 12 2017

Keywords

Comments

a(n) is the n-th term of the main diagonal of iterated partial sums array of powers of n (see example).

Examples

			For n = 2 we have:
----------------------------
0   1   [2]   3    4     5
----------------------------
1,  2,   4,   8,  16,   32, ... A000079 (powers of 2)
1,  3,   7,  15,  31,   63, ... A126646 (partial sums of A000079)
1,  4, [11], 26,  57,  120, ... A000295 (partial sums of A126646)
----------------------------
therefore a(2) = 11.

Crossrefs

Cf. A000312, A001700, A031973, A032443, A100192, A100193.

Programs

  • Mathematica
    Join[{1}, Table[Sum[n^(n - k) Binomial[n + k - 1, k], {k, 0, n}], {n, 1, 20}]]
Table[SeriesCoefficient[1/((1 - x)^n (1 - n x)), {x, 0, n}], {n, 0, 20}]
Join[{1, 2}, Table[n^(2 n)/(n - 1)^n - Binomial[2 n, n + 1] Hypergeometric2F1[1, 2 n + 1, n + 2, 1/n]/n, {n, 2, 20}]]
  • PARI
    a(n) = sum(k=0, n, n^(n-k)*binomial(n+k-1,k)); \\ Michel Marcus, Oct 12 2017

Formula

a(n) = [x^n] 1/((1 - x)^n*(1 - n*x)).
a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Oct 16 2017

A384365 a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 9, 67, 458, 2979, 18750, 115278, 696372, 4149283, 24452534, 142808922, 827780684, 4767638158, 27309438252, 155689424316, 883891633896, 4999703023395, 28188457323366, 158463492162594, 888473780483292, 4969653746436762, 27737520941131140, 154507945286680452
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A100193, A386940, A386941.
Cf. A088218, A258431, A386955, A386956.

Programs

  • Magma
    [&+[ (k+1) * 3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[(k+1) * 3^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, (k+1)*3^k*binomial(2*n+1, n-k));

Formula

a(n) = [x^n] 1/((1-4*x)^2 * (1-x)^n).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} (k+1) * 4^k * binomial(2*n-k-1,n-k).
G.f.: (1+sqrt(1-4*x))/( 2 * sqrt(1-4*x) * (2*sqrt(1-4*x)-1)^2 ).
D-finite with recurrence +27*n*a(n) +6*(-58*n+17)*a(n-1) +32*(46*n-37)*a(n-2) +1024*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Aug 19 2025
a(n) ~ n * 2^(4*n+1) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025

A386940 a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 3, 13, 60, 285, 1378, 6748, 33372, 166365, 834900, 4213638, 21368724, 108820764, 556184580, 2851679620, 14661848560, 75568345821, 390330333402, 2020046912260, 10472193542100, 54373036935910, 282704274266040, 1471722678992700, 7670327017789800, 40017679829372700
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2025

Keywords

Crossrefs

Cf. A100193, A384365, A386941.
Cf. A000984, A293490.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(2*k, k)*binomial(2*n-k-1, n-k));

Formula

a(n) = [x^n] 1/(sqrt(1-4*x) * (1-x)^n).
G.f.: (1+sqrt(1-4*x))/sqrt( 4 * (1-4*x) * (2*sqrt(1-4*x)-1) ).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n-1/2,k) * binomial(n-k-1/2,n-k) = Sum_{k=0..n} (3/4)^k * binomial(2*k,k) * binomial(2*n-1/2,n-k).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n-1/2,k) * binomial(2*n-k-1,n-k).

A386941 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 7, 45, 276, 1645, 9618, 55468, 316620, 1792989, 10089420, 56482998, 314859636, 1748876220, 9684449908, 53487036420, 294732771280, 1620825793053, 8897604701130, 48766676365204, 266905699036900, 1458941915879910, 7965552023094600, 43444688665988700
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2025

Keywords

Crossrefs

Cf. A100193, A384365, A386940.

Programs

  • PARI
    a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*n-k-1, n-k));

Formula

a(n) = [x^n] 1/((1-4*x)^(3/2) * (1-x)^n).
G.f.: (1+sqrt(1-4*x))/sqrt( 4 * (1-4*x) * (2*sqrt(1-4*x)-1)^3 ).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+1/2,k) * binomial(n-k+1/2,n-k) = Sum_{k=0..n} (2*k+1) * (3/4)^k * binomial(2*k,k) * binomial(2*n+1/2,n-k).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1/2,k) * binomial(2*n-k-1,n-k).

A386960 a(n) = Sum_{k=0..n} 8^k * binomial(2*n,n-k).

Original entry on oeis.org

1, 10, 102, 1036, 10502, 106380, 1077276, 10908096, 110447046, 1118286172, 11322685172, 114642332232, 1160754172316, 11752638152824, 118995469654968, 1204829162684136, 12198895398209862, 123513816397462524, 1250577392936568708, 12662096110945862856, 128203723152486704052
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A032443, A072547, A088218, A100192, A100193.

Programs

  • Magma
    [&+[8^k * Binomial(2*n, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 13 2025
  • Mathematica
    Table[Sum[8^k*Binomial[2*n,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 13 2025 *)
  • PARI
    a(n) = sum(k=0, n, 8^k*binomial(2*n, n-k));

Formula

a(n) = [x^n] 1/((1-9*x) * (1-x)^n).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k-1,n-k).
G.f.: (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).
Showing 1-6 of 6 results.

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