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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A367113 G.f. satisfies A(x) = 1 + 2*x*A(x) + 2*x^3*A(x)^3.

Original entry on oeis.org

1, 2, 4, 10, 32, 112, 396, 1416, 5184, 19424, 73984, 285056, 1108848, 4350816, 17203008, 68473504, 274122752, 1103014912, 4458611968, 18096793088, 73724852224, 301360575488, 1235633545216, 5080554352640, 20943623880448, 86541514460672, 358386391051264
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2023

Keywords

Crossrefs

Cf. A001764, A071879, A367111, A367112.
Cf. A071356, A367115.
Cf. A367074.

Programs

  • PARI
    a(n) = sum(k=0, n\3, 2^(n-2*k)*binomial(n, 3*k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(n,3*k) * A001764(k).

A100238 G.f. A(x) satisfies: 2^n + 1 = Sum_{k=0..n} [x^k] A(x)^n for n>=1.

Original entry on oeis.org

1, 2, -2, 4, -12, 40, -144, 544, -2128, 8544, -35008, 145792, -615296, 2625792, -11311616, 49124352, -214838528, 945350144, -4182412288, 18593224704, -83015133184, 372090122240, -1673660915712, 7552262979584, -34178799378432, 155096251351040, -705533929816064
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2004

Keywords

Examples

			From the table of powers of A(x), we see that
2^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1: [1, 2], -2, 4, -12, 40, -144, 544, -2128, 8544, ...;
A^2: [1, 4, 0], 0, -4, 16, -64, 256, -1040, 4288, ...;
A^3: [1, 6, 6, -4], 0, 0, -8, 48, -240, 1120, -5088, ...;
A^4: [1, 8, 16, 0, -8], 0, 0, 0, -16, 128, -768, ...;
A^5: [1, 10, 30, 20, -20, -8], 0, 0, 0, 0, -32, ...;
A^6: [1, 12, 48, 64, -12, -48, 0], 0, 0, 0, 0, 0, ...;
A^7: [1, 14, 70, 140, 56, -112, -56, 16], 0, 0, 0, ...;
A^8: [1, 16, 96, 256, 240, -128, -256, 0, 32], 0, 0, ...; ...
In the above table of coefficients in A(x)^n, the main diagonal satisfies:
[x^n] A(x)^(n+1) = (n+1)*A009545(n+1) for n>=0.

Crossrefs

Cf. A100223, A071356, A009545, A143045.
a(n) = -(-1)^n * A025227(n), if n>1.

Programs

  • PARI
    {a(n)=if(n==0,1,(2^n+1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)}
  • PARI
    {a(n)=if(n==0,1,if(n==1,2,if(n==2,-2,(-2*(2*n-3)*a(n-1)+4*(n-3)*a(n-2))/n)))}
  • PARI
    {a(n)=polcoeff( (1+2*x+sqrt(1+4*x-4*x^2+x^2*O(x^n)))/2,n)}
  • PARI
    a(n)=polcoeff((1+2*x+sqrt(1+4*x-4*x^2+x*O(x^n)))/2,n)

Formula

a(n) = (-2*(2*n-3)*a(n-1) + 4*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=2, a(2)=-2.
G.f.: A(x) = (1+2*x + sqrt(1+4*x-4*x^2))/2.
G.f. satisfies: (2+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k] (A(x)+z*x)^n for all z, where [x^k] F(x) denotes the coefficient of x^k in F(x).
Given g.f. A(x), then B(x)=A(x)-1-x series reversion is -B(-x). - Michael Somos, Sep 07 2005
Given g.f. A(x) and C(x) = g.f. of A025225, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - Michael Somos, Sep 07 2005
G.f.: 4x^2/(1+2x - sqrt(1+4x-4x^2)). - Michael Somos, Sep 08 2005

A136576 Series reversion of x*c(x)/(1 - 2*x), c(x) the g.f. of A000108.

Original entry on oeis.org

0, 1, -3, 10, -36, 136, -532, 2136, -8752, 36448, -153824, 656448, -2827904, 12281088, -53709632, 236337536, -1045603072, 4648306176, -20753783296, 93022530560, -418415228928, 1888065744896, -8544699844608, 38774062837760
Offset: 0

Views

Author

Paul Barry, Jan 08 2008

Keywords

Comments

Hankel transform of a(n+1) is A136577 (conjecture).

Links

Crossrefs

Cf. A069731, A071356.

Programs

  • Mathematica
    CoefficientList[Series[(Sqrt[1+4*x-4*x^2]+4*x^2-2*x-1)/(8*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 22 2014 *)
  • PARI
    x='x+O('x^50); concat([0], Vec((sqrt(1+4*x-4*x^2)+4*x^2-2*x-1)/(8*x^2))) \\ G. C. Greubel, Mar 21 2017

Formula

G.f.: (sqrt(1+4*x-4*x^2)+4*x^2-2*x-1)/(8*x^2).
D-finite with recurrence (n+2)*a(n) + 2*(2*n+1)*a(n-1) + 4*(1-n)*a(n-2) = 0. - R. J. Mathar, Dec 11 2011
a(n) ~ (-1)^(n+1) * (3+2*sqrt(2)) * sqrt(4-2*sqrt(2)) * 2^(n-2) * (1+sqrt(2))^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 22 2014
For n >= 1, a(n) = (-1)^(n+1) * (1/2) * A071356(n) = (-1)^(n+1) * Sum_{k = 0..floor(n/2)} binomial(n, 2*k)*Catalan(k)*2^(n-k-1). The recurrence given above follows from this using the WZ algorithm. - Peter Bala, Apr 28 2024

A179190 Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4*x - 4*x^2).

Original entry on oeis.org

1, 2, 4, 8, 24, 80, 288, 1088, 4256, 17088, 70016, 291584, 1230592, 5251584, 22623232, 98248704, 429677056, 1890700288, 8364824576, 37186449408, 166030266368, 744180244480, 3347321831424, 15104525959168, 68357598756864
Offset: 0

Views

Author

Clark Kimberling, Jul 01 2010

Keywords

Examples

			The Maclaurin series is 1 + 2*x + 4*x^2 + 8*x^3 + 24*x^4 + ...

Links

Crossrefs

Cf. A178693, A178694, A179191.

Programs

  • Maple
    A179190 := proc(n) if n = 0 then 1; else add( doublefactorial(2*n-2*k-3) *2^(n-k) / k! / (n-2*k)!, k=0..floor(n/2)) ; end if; end proc: # R. J. Mathar, Jul 11 2011
  • Mathematica
    Table[SeriesCoefficient[Series[2-Sqrt[1-4*t-4*t^2], {t,0,n}], n], {n, 0, 30}] (* G. C. Greubel, Jan 25 2019 *)
  • Maxima
    makelist(coeff(taylor(2-sqrt(1-4*x-4*x^2), x, 0, n), x, n), n, 0, 24); /* Bruno Berselli, Jul 04 2011 */

Formula

G.f.: 2 - sqrt(1 - 4*x - 4*x^2).
a(n) = 4*A071356(n-2), n >= 2. - R. J. Mathar, Jul 08 2010
a(n) = Sum_{k=0..floor(n/2)} (2*n - 2*k - 3)!! *2^(n-k)/(k!*(n-2k)!), n > 0. - R. J. Mathar, Jul 11 2011
a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 26 2019
D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1) +4*(-n+3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

A307969 Coefficient of x^n in 1/(n+1) * (1 - 2*x - 2*x^2)^(n+1).

Original entry on oeis.org

1, -2, 2, 4, -24, 48, 24, -464, 1376, -704, -9920, 41600, -55040, -201216, 1266048, -2628864, -3021312, 37696512, -108659712, 15857664, 1067003904, -4155138048, 4378226688, 27416125440, -149814263808, 273526325248, 569660309504, -5103546466304, 13241145229312, 4914079858688
Offset: 0

Views

Author

Seiichi Manyama, May 08 2019

Keywords

Comments

Also coefficient of x^n in the expansion of 2/(1 + 2*x + sqrt(1 + 4*x + 12*x^2)).

Links

Crossrefs

Column 2 of A307968.
Cf. A000108, A071356, A116093.

Programs

  • Mathematica
    a[n_] := Sum[(-2)^(n-k) * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Table[(-2)^n * Hypergeometric2F1[1/2 - n/2, -n/2, 2, -2], {n, 0, 30}] (* Vaclav Kotesovec, May 12 2021 *)
  • PARI
    {a(n) = polcoef((1-2*x-2*x^2)^(n+1)/(n+1), n)}
  • PARI
    {a(n) = sum(k=0, n\2, (-2)^(n-k)*binomial(n, k)*binomial(n-k, k)/(k+1))}
  • PARI
    {a(n) = sum(k=0, n\2, (-2)^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1))}

Formula

a(n) = Sum_{k=0..floor(n/2)} (-2)^(n-k) * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-2)^(n-k) * binomial(n,2*k) * A000108(k).
(n+2) * a(n) = -2 * (2*n+1) * a(n-1) - 12 * (n-1) * a(n-2).
a(n) = (-2)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 2, -2). - Vaclav Kotesovec, May 12 2021

A369128 Expansion of (1/x) * Series_Reversion( x / ((1+x)^5+x^5) ).

Original entry on oeis.org

1, 5, 35, 285, 2530, 23752, 231910, 2331040, 23960235, 250692365, 2661086895, 28587333725, 310217791590, 3395464391870, 37442295427120, 415570885425280, 4638842010800025, 52044582325415025, 586553425250933055, 6637525235622842585, 75387741117556006435
Offset: 0

Views

Author

Seiichi Manyama, Jan 14 2024

Keywords

Links

Crossrefs

Cf. A071356, A192132, A369126.
Cf. A364522.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^5+x^5))/x)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+1, k)*binomial(5*n-5*k+5, n-5*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+1,k) * binomial(5*n-5*k+5,n-5*k).

A369212 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^3) ).

Original entry on oeis.org

1, 2, 5, 15, 50, 177, 652, 2473, 9594, 37892, 151846, 615859, 2523217, 10427471, 43415259, 181941198, 766841846, 3248517320, 13823977350, 59067577266, 253315964424, 1089998388418, 4704475230340, 20361365646315, 88351705071583, 384280788724692, 1675063399090659
Offset: 0

Views

Author

Seiichi Manyama, Jan 16 2024

Keywords

Links

Crossrefs

Cf. A002212, A071356, A369158.
Cf. A071879, A192132.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^3))/x)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n+1, k)*binomial(2*n-2*k+2, n-3*k))/(n+1);

Formula

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

A367115 G.f. satisfies A(x) = 1 + 2*x*A(x) + 2*x^4*A(x)^4.

Original entry on oeis.org

1, 2, 4, 8, 18, 52, 184, 688, 2512, 8864, 30784, 107648, 384432, 1403872, 5205568, 19443328, 72817856, 273199488, 1027939072, 3883718144, 14741042464, 56189409088, 214931447680, 824443822848, 3169934397184, 12214858010112, 47168251137024
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2023

Keywords

Crossrefs

Cf. A002293, A127902, A190590, A367114.
Cf. A071356, A367113.

Programs

  • PARI
    a(n) = sum(k=0, n\4, 2^(n-3*k)*binomial(n, 4*k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} 2^(n-3*k) * binomial(n,4*k) * A002293(k).

A371714 Expansion of g.f. A(x) satisfying A( x^3*A(x) - x^3*A(x)^2 ) = x^4.

Original entry on oeis.org

1, 1, 2, 5, 13, 40, 126, 409, 1360, 4611, 15878, 55384, 195282, 694910, 2492454, 9001405, 32704855, 119462142, 438441266, 1616001547, 5979144981, 22199682130, 82685478702, 308864831632, 1156806962608, 4343254831180, 16343719170558, 61630500821158, 232854921227616, 881378279895534
Offset: 1

Views

Author

Paul D. Hanna, Apr 04 2024

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 40*x^6 + 126*x^7 + 409*x^8 + 1360*x^9 + 4611*x^10 + 15878*x^11 + 55384*x^12 + ...
where A( x^3*A(x)*(1 - A(x)) ) = x^4.
RELATED SERIES.
(1) Let R(x) be the series reversion of A(x), R(A(x)) = x, where
R(x) = x - x^2 + x^5 - 5*x^6 + 10*x^7 - 10*x^8 + 10*x^9 - 46*x^10 + 180*x^11 - 420*x^12 + 665*x^13 - 1085*x^14 + 3150*x^15 - 10190*x^16 + ...
upon comparing the expansion of R(x) to the series
A(x)*(1 - A(x)) = x - x^5 + x^17 - 5*x^21 + 10*x^25 - 10*x^29 + 10*x^33 - 46*x^37 + 180*x^41 - 420*x^45 + 665*x^49 - 1085*x^53 + ...
we see that A(x)*(1 - A(x)) = R(x^4)/x^3.
(2) Let B(x) be the even bisection of A(x),
B(x) = x^2 + 5*x^4 + 40*x^6 + 409*x^8 + 4611*x^10 + 55384*x^12 + ...,
then
B( x^3*A(x)*(1 - A(x)) ) = x^8 - 2*x^12 + 6*x^16 - 20*x^20 + 72*x^24 - 272*x^28 + 1064*x^32 + ... + (-1)^(n-1)*A071356(n-1)*x^(8*n) + ...
that is,
B( x^3*A(x)*(1 - A(x)) ) = (1 + 2*x^4 - sqrt(1 + 4*x^4 - 4*x^8))/4.
SPECIFIC VALUES.
Let r be the radius of convergence, then A(r) = 1/2, and
r = A(r^3/4)^(1/4) = 0.2509961746510523531562794924202947105158...
A(1/4) = 0.4687500009132742494908083392815082722109...
A(1/5) = 0.2756788017179389881387593924191299703438...
A(1/6) = 0.2111022081144963995053917910635203605728...
A(1/8) = 0.1464034536677082355575260818928469620931...

Links

Crossrefs

Cf. A291614, A272463, A071356.

Programs

  • PARI
    {a(n) = my(A=[1, 1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff(x^4 - subst(F, x, x^3*F - x^3*F^2), #A+3) ); A[n]}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A( x^3*A(x)*(1 - A(x)) ) = x^4.
(2) A( -x^3*A(x)*(1 - A(x)) ) = (1 - sqrt(1 + 4*x^4 - 4*x^8))/2.
(3) A(x) = (1 - sqrt(1 - 4*R(x^4)/x^3))/2, where R(A(x)) = x.
a(n) ~ c * d^n / n^(3/2), where d = 3.9841244648060905977016688650241255776651... and c = 0.13991881826475367145488117165180720475565183... - Vaclav Kotesovec, Apr 05 2024
Let r be the radius of convergence, then A(r) = 1/2, where r = A(r^3/4)^(1/4) = 0.2509961746510523531... = 1/d (d is given above). - Paul D. Hanna, Apr 06 2024

A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.

Original entry on oeis.org

1, 4, 18, 80, 360, 1632, 7448, 34176, 157536, 728960, 3384128, 15754752, 73525504, 343870464, 1611288960, 7562801152, 35550504448, 167339022336, 788643765248, 3720901222400, 17573439614976, 83074892775424, 393056192851968, 1861155016212480, 8819174122700800, 41818448615636992
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2025

Keywords

Links

Crossrefs

Cf. A006139, A387402, A387403.
Cf. A071356, A374497.

Programs

  • Magma
    [&+[2^(n-k) * Binomial(n+1,n-2*k) * Binomial(2*k+1,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
  • Mathematica
    Table[Sum[2^(n-k)*Binomial[n+1,n-2*k]*Binomial[2*k+1,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+1, n-2*k)*binomial(2*k+1, k));

Formula

n*(n+2)*a(n) = (n+1) * (2*(2*n+1)*a(n-1) + 4*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+1).
E.g.f.: exp(2*x) * BesselI(1, 2*sqrt(2)*x) / sqrt(2), with offset 1.
a(n) = (n+1) * A071356(n).
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