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.

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A365085 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x))^2.

Original entry on oeis.org

1, 1, -1, -2, 5, 6, -30, -13, 189, -56, -1188, 1266, 7194, -14377, -40183, 135278, 188773, -1151800, -503880, 9109076, -3419924, -67220176, 80390824, 458183898, -998680470, -2794491329, 10156144385, 13919066170, -92250872385, -36047778330, 769826420850, -339940775445
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Crossrefs

Cf. A090192, A365086, A365087, A365088.
Cf. A106510, A364735.

Programs

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

Formula

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

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).

A365087 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x))^4.

Original entry on oeis.org

1, 1, -3, -1, 29, -44, -265, 1114, 1369, -19076, 20388, 250977, -875281, -2116594, 19136754, -7765108, -306092007, 830209808, 3388957208, -22266676364, -8185922076, 413223401045, -814031607979, -5513566634947, 27558060911119, 35395095404776
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Crossrefs

Cf. A090192, A365085, A365086, A365088.
Cf. A321798, A364737, A365083.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(n+3*k-1, n-k)/(n-k+1));

Formula

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

A365088 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x))^5.

Original entry on oeis.org

1, 1, -4, 1, 46, -129, -405, 3319, -1617, -59258, 199541, 642170, -6038395, 3886091, 119884973, -440626784, -1367688245, 14055527190, -11043763380, -290488387366, 1137260033731, 3336325340735, -36966844508130, 34098313310315, 776097820004580
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Crossrefs

Cf. A090192, A365085, A365086, A365087.
Cf. A321799, A364738, A365084.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(n+4*k-1, n-k)/(n-k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(n+4*k-1,n-k) / (n-k+1).

A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 20, 16, 6, 1, 0, 70, 64, 30, 8, 1, 0, 252, 256, 140, 48, 10, 1, 0, 924, 1024, 630, 256, 70, 12, 1, 0, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 0, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 01 2012

Keywords

Comments

Riordan array (1, x/sqrt(1-4*x)). Inverse of Riordan array (1, x*exp(arcsinh(-2*x))).
T is the convolution triangle of the shifted central binomial coefficients binomial(2*(n-1), n-1). - Peter Luschny, Oct 19 2022

Examples

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   6,   4,   1;
  0,  20,  16,   6,   1;
  0,  70,  64,  30,   8,   1;
  0, 252, 256, 140,  48,  10,   1;

Crossrefs

Cf. A054335 and columns listed there.

Programs

  • Maple
    # Uses function PMatrix from A357368.
PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022

Formula

T(n,n) = 1 = A000012(n); T(n+1,n) = 2*n = A005843(n); T(n+2,n) = 2*n*(n+2) = A054000(n+1).
Sum_{k=0..n} T(n,k)*x^k = -A081696(n-1), A000007(n), A026671(n-1), A084868(n) for x = -1, 0, 1, 2 respectively.
G.f.: sqrt(1-4*x)/(sqrt(1-4*x)-y*x).
Sum_{k=0..n} T(n,k)*A090192(k) = A000108(n), A000108 = Catalan numbers.

A210628 Expansion of (-1 + 2*x + sqrt( 1 - 4*x^2)) / (2*x) in powers of x.

Original entry on oeis.org

1, -1, 0, -1, 0, -2, 0, -5, 0, -14, 0, -42, 0, -132, 0, -429, 0, -1430, 0, -4862, 0, -16796, 0, -58786, 0, -208012, 0, -742900, 0, -2674440, 0, -9694845, 0, -35357670, 0, -129644790, 0, -477638700, 0, -1767263190, 0, -6564120420, 0, -24466267020, 0
Offset: 0

Views

Author

Michael Somos, Mar 25 2012

Keywords

Comments

Except for the leading term, the sequence is equal to -A097331(n). - Fung Lam, Mar 22 2014

Examples

			G.f. = 1 - x - x^3 - 2*x^5 - 5*x^7 - 14*x^9 - 42*x^11 - 132*x^13 - 429*x^15 + ...

Links

Crossrefs

Cf. A000108, A001405, A090192, A097331, A126930, A210736.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 + 2*x + Sqrt(1-4*x^2))/(2*x))); // G. C. Greubel, Aug 11 2018
  • Mathematica
    CoefficientList[Series[1 - 2 x/(1 + Sqrt[1 - 4 x^2]), {x, 0, 45}], x] (* Bruno Berselli, Mar 25 2012 *)
a[ n_] := SeriesCoefficient[ (-1 + 2 x + Sqrt[1 - 4 x^2]) / (2 x), {x, 0, n}];
  • Maxima
    makelist(coeff(taylor(1-2*x/(1+sqrt(1-4*x^2)), x, 0, n), x, n), n, 0, 45); /* Bruno Berselli, Mar 25 2012 */
  • PARI
    {a(n) = polcoeff( (-1 + 2*x + sqrt( 1 - 4*x^2 + x^2 * O(x^n))) / (2*x), n)};
  • PARI
    {a(n) = if( n<1, n==0, polcoeff( serreverse( -x / (1 + x^2) + x * O(x^n)), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 - x - x * (1 - A)^2); polcoeff( A, n))};

Formula

G.f.: 1 - (2*x) / (1 + sqrt( 1 - 4*x^2)) = 1 - (1 - sqrt( 1 - 4*x^2)) / (2*x).
G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = x*y^2 - (1 - 2*x) * (1 - y).
G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 - x.
G.f. A(x) = 1 - x - x * (1 - A(x))^2 = 1 - 1/x + 1 / (1 - A(x)).
G.f. A(x) = 1 / (1 + x / (1 - 2*x + x * A(x))).
G.f. A(x) = 1 / (1 + x / (1 - x / (1 - x / (1 + x * A(x))))).
G.f. A(x) = 1 / (1 + x * A001405(x)). A126930(x) = 1 / (1 + x * A(x)).
G.f. A(x) = 1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...))). - Michael Somos, Jan 02 2013
a(2*n) = 0 unless n=0, a(2*n + 1) = -A000108(n). a(n) = (-1)^n * A097331(n). a(n-1) = (-1)^floor(n/2) * A090192(n).
Convolution inverse of A210736. - Michael Somos, Jan 02 2013
G.f.: 2/( G(0) + 1), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1+2*x) - 2*x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+2*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
D-finite with recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 17 2014
For nonzero odd-power terms, a(n) = -2^(n+1)/(n+1)^(3/2)/sqrt(2*Pi)*(1+3/(4*n) + O(1/n^2)). (with contribution of Vaclav Kotesovec) - Fung Lam, Mar 17 2014

A376134 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, -1, -6, 17, 141, -660, -6688, 43837, 521755, -4412893, -60477282, 628119268, 9772644140, -120524236108, -2103803950976, 30068650440341, 582807287964375, -9477098158324107, -202143447363632090, 3686281848172281145, 85853256990102196221, -1735552985238117874788
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2024

Keywords

Crossrefs

Cf. A088716, A090192, A105523, A376135, A376137.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1/(1 - x A[-x] + x^2 A'[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(-x) + x^2 * A'(-x)).

A376135 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (2*k+1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, -2, -15, 86, 1030, -9844, -156219, 2098406, 41282298, -716119260, -16837011158, 358425572604, 9820300812556, -247923816153128, -7765514675946195, 226869417798485382, 8001626352728559218, -265582398152349968716, -10419379442081103988738
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2024

Keywords

Crossrefs

Cf. A000699, A090192, A105523, A376134, A376137.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (2 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 - x A[-x] + 2 x^2 A'[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(-x) + 2 * x^2 * A'(-x)).

A376137 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1)^2 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, -3, -34, 495, 13631, -467404, -23984426, 1490938299, 123999435015, -12164649041259, -1497474725212924, 212746558833692052, 36393896155519042476, -7062273474686464802160, -1603475573855830444120802, 407344895625777134555939139, 118552169162473363108837155199, -38177398083353809033748641523305
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2024

Keywords

Crossrefs

Cf. A090192, A105523, A376095, A376134, A376135.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 0; Do[A[x] = 1/(1 - x A[-x] + 3 x^2 A'[-x] - x^3 A''[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(-x) + 3 * x^2 * A'(-x) - x^3 * A''(-x)).

A123254 Triangle T(n,k), 0<=k<=n, read by rows given by [ -1,1,-1,1,-1,1,-1,1,-1,1,...] DELTA [1,-1,1,-1,1,-1,1,-1,1,-1,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, 0, 0, 1, -3, 3, -1, 0, 0, 0, 0, 0, -2, 10, -20, 20, -10, 2, 0, 0, 0, 0, 0, 0, 0, 5, -35, 105, -175, 175, -105, 35, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 126, -504, 1176, -1764, 1764, -1176, 504, -126, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Oct 08 2006

Keywords

Examples

			Triangle begins:
1;
-1, 1;
0, 0, 0;
1, -3, 3, -1;
0, 0, 0, 0, 0;
-2, 10, -20, 20, -10, 2;
0, 0, 0, 0, 0, 0, 0;
5, -35, 105, -175, 175, -106, 35, -5;
0, 0, 0, 0, 0, 0, 0, 0, 0;

Crossrefs

Cf. A000108, A007318.

Formula

T(n,k)=(-1)^k*A105523(n)*binomial(n,k) . T(n,n)=A090192(n) . Sum_{k, 0<=k<=n}T(n,k)= 0^n= A000007(n).
Previous Showing 31-40 of 41 results. Next

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