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.

A127897 Series reversion of x/(1 + 2*x + 3*x^2 + x^3).

Original entry on oeis.org

0, 1, 2, 7, 27, 114, 507, 2342, 11125, 54002, 266684, 1335610, 6767477, 34629709, 178701317, 928903447, 4859345882, 25563551782, 135153617840, 717740916202, 3826894116962, 20478451476328, 109945087353190, 592048943478464, 3196930550222605, 17306392059508743, 93905862139673832
Offset: 0

Views

Author

Paul Barry, Feb 04 2007

Keywords

Comments

Series reversion of A127896.

Links

Programs

  • Mathematica
    Flatten[{0,Rest[CoefficientList[Series[2*Sqrt[3]*Sqrt[(1+x)/x]*Sin[ArcSin[3*Sqrt[3]/(2*Sqrt[(1+x)/x])]/3]/3, {x, 0, 20}], x]]}] (* Vaclav Kotesovec, Oct 20 2012 *)
  • PARI
    {a(n) = my(A = sum(m=1,n, binomial(3*m, m-1)/m * x^m / (1+x +x*O(x^n))^m ) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Feb 04 2018

Formula

G.f.: 2*sqrt(3)*sqrt((1+x)/x)*sin(arcsin(3*sqrt(3)/(2*sqrt((1+x)/x)))/3)/3;
a(n) = Sum_{k=0..n-1} Sum_{j=0..k} (1/(2k+j-1))*C(n-1,3k-j)*C(3k-j,k)*C(k,j)*2^(n-3k+j-1)*3^j;
Recurrence: 2*n*(2*n+1)*a(n) = (3*n-1)*(5*n-2)*a(n-1) + 2*(n-2)*(21*n-20)*a(n-2) + 23*(n-3)*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 23^(n+1/2)/(12*4^n*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
G.f.: Sum_{n>=1} binomial(3*n, n-1)/n * x^n / (1+x)^n. - Paul D. Hanna, Feb 04 2018
G.f. A(x) satisfies: A(x) = x * (1 + 2*A(x) + 3*A(x)^2 + A(x)^3). - Ilya Gutkovskiy, Jul 01 2020

A365086 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x))^3.

Original entry on oeis.org

1, 1, -2, -2, 15, -4, -122, 204, 903, -3374, -4635, 43539, -13233, -475123, 873392, 4244591, -16906773, -24952174, 244162840, -74520792, -2901715074, 5483226036, 27740164293, -112969486284, -172903931727, 1714556657881, -513739179725, -21235809823325
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Crossrefs

Cf. A090192, A365085, A365087, A365088.
Cf. A127896, A161797, A364736.

Programs

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

Formula

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

A127895 Riordan array (1/(1+x)^3, x/(1+x)^3).

Original entry on oeis.org

1, -3, 1, 6, -6, 1, -10, 21, -9, 1, 15, -56, 45, -12, 1, -21, 126, -165, 78, -15, 1, 28, -252, 495, -364, 120, -18, 1, -36, 462, -1287, 1365, -680, 171, -21, 1, 45, -792, 3003, -4368, 3060, -1140, 231, -24, 1, -55, 1287, -6435, 12376, -11628, 5985, -1771, 300, -27, 1
Offset: 0

Views

Author

Paul Barry, Feb 04 2007

Keywords

Comments

The matrix inverse of the convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022

Examples

			Triangle begins
    1;
   -3,     1;
    6,    -6,     1;
  -10,    21,    -9,      1;
   15,   -56,    45,    -12,      1;
  -21,   126,  -165,     78,    -15,      1;
   28,  -252,   495,   -364,    120,    -18,     1;
  -36,   462, -1287,   1365,   -680,    171,   -21,     1;
   45,  -792,  3003,  -4368,   3060,  -1140,   231,   -24,   1;
  -55,  1287, -6435,  12376, -11628,   5985, -1771,   300, -27,   1;
   66, -2002, 12870, -31824,  38760, -26334, 10626, -2600, 378, -30, 1;

Links

Crossrefs

Inverse is A127898.
Alternating sign version of A127893.
Cf. A001764, A095263, A127896.

Programs

  • Magma
    [(-1)^(n-k)*Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Apr 29 2018
  • Maple
    # Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022
  • Mathematica
    Table[(-1)^(n-k)*Binomial[n+2*k+2, n-k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 29 2018 *)
  • PARI
    for(n=0, 10, for(k=0,n, print1((-1)^(n-k)*binomial(n+2*k+2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018
  • Sage
    flatten([[(-1)^(n-k)*binomial(n+2*k+2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

Formula

T(n, k) = (-1)^(n-k)*binomial(n +2*k +2, n-k).
Sum_{k=0..n} T(n, k) = A127896(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A095263(n) (diagonal sums).

Extensions

Terms a(50) onward added by G. C. Greubel, Apr 29 2018

A365083 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x)^4.

Original entry on oeis.org

1, 1, -3, 3, 5, -22, 27, 28, -163, 235, 134, -1188, 1983, 408, -8504, 16320, -1551, -59659, 131507, -46683, -408806, 1040147, -612380, -2721835, 8088003, -6523626, -17457420, 61883839, -62900496, -106248240, 466069760, -571001695, -595520019, 3454539427
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Links

Crossrefs

Cf. A106510, A127896, A365084.
Cf. A055991.

Programs

  • Mathematica
    LinearRecurrence[{-3, -6, -4, -1}, {1, 1, -3, 3, 5}, 1 + 33] (* Robert P. P. McKone, Aug 21 2023 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n+3*k-1, n-k));

Formula

G.f.: A(x) = 1/( 1 - x/(1+x)^4 ).
a(n) = -3*a(n-1) - 6*a(n-2) - 4*a(n-3) - a(n-4) for n > 4.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+3*k-1,n-k).

A365084 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x)^5.

Original entry on oeis.org

1, 1, -4, 6, 6, -49, 95, 24, -592, 1417, -414, -6809, 20142, -14831, -73353, 274761, -311105, -715647, 3607624, -5463428, -5785294, 45588556, -87189477, -25565196, 552659892, -1305250324, 340413165, 6379267117, -18606431142, 13202513476, 69064770845
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Links

Crossrefs

Cf. A106510, A127896, A365083.
Cf. A079675.

Programs

  • Mathematica
    LinearRecurrence[{-4, -10, -10, -5, -1}, {1, 1, -4, 6, 6, -49}, 1 + 30] (* Robert P. P. McKone, Aug 21 2023 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n+4*k-1, n-k));

Formula

G.f.: A(x) = 1/( 1 - x/(1+x)^5 ).
a(n) = -4*a(n-1) - 10*a(n-2) - 10*a(n-3) - 5*a(n-4) - a(n-5) for n > 5.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+4*k-1,n-k).

A233581 a(n) = 2*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 1, a(1) = 0, a(2) = -1.

Original entry on oeis.org

1, 0, -1, -1, 1, 4, 4, -3, -14, -15, 9, 49, 56, -26, -171, -208, 71, 595, 769, -176, -2064, -2831, 354, 7137, 10381, -295, -24596, -37926, -2359, 84464, 138079, 20407, -288959, -501060, -114836, 984549, 1812546, 556609, -3339871, -6537023, -2497824, 11275550
Offset: 0

Views

Author

Michael Somos, Dec 14 2013

Keywords

Examples

			G.f. = 1 - x^2 - x^3 + x^4 + 4*x^5 + 4*x^6 - 3*x^7 - 14*x^8 - 15*x^9 + ...

Links

Crossrefs

Cf. A034943, A052921.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+2*x^2)/(1-2*x+3*x^2-x^3))); // G. C. Greubel, Aug 08 2018
  • Mathematica
    CoefficientList[Series[(1-2*x+2*x^2)/(1-2*x+3*x^2-x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{2,-3,1}, {1,0,-1}, 50] (* G. C. Greubel, Aug 08 2018 *)
  • PARI
    {a(n) = if( n<0, polcoeff( (1 - x) / (1 - 3*x + 2*x^2 - x^3) + x * O(x^-n), -n), polcoeff( (1 - 2*x + 2*x^2) / (1 - 2*x + 3*x^2 - x^3) + x * O(x^n), n))}

Formula

G.f.: (1 - 2*x + 2*x^2) / (1 - 2*x + 3*x^2 - x^3).
a(n) = A052921(-n). a(n)^2 - a(n-1)*a(n+1) = A034943(n).
a(n) = A127896(n) -2*A127896(n-1) + 2*A127896(n-2). - R. J. Mathar, Sep 24 2021
Showing 1-6 of 6 results.

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