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.

Previous Showing 11-18 of 18 results.

A307528 G.f. A(x) satisfies: A(x) = 1 + x^2*A(x)^2/(1 - x*A(x) - x^2*A(x)^2 - x^3*A(x)^3).

Original entry on oeis.org

1, 0, 1, 1, 4, 9, 27, 76, 226, 680, 2078, 6441, 20153, 63684, 202732, 649930, 2095854, 6794684, 22131765, 72393439, 237703654, 783198068, 2588645047, 8580674778, 28517805357, 95009277576, 317242351135, 1061500510809, 3558683892258, 11952025977378, 40209157279701
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2019

Keywords

Examples

			G.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 9*x^5 + 27*x^6 + 76*x^7 + 226*x^8 + 680*x^9 + 2078*x^10 + ...

Crossrefs

Cf. A000073, A049124, A049140, A307411, A307412, A307413, A307529.

Programs

  • Mathematica
    terms = 31; CoefficientList[1/x InverseSeries[Series[x (1 - x - x^2 - x^3)/(1 - x - x^3), {x, 0, terms}], x], x]
terms = 30; A[] = 0; Do[A[x] = 1 + x^2 A[x]^2/(1 - x A[x] - x^2 A[x]^2 - x^3 A[x]^3) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 31; t[n_] := t[n] = SeriesCoefficient[x^2/(1 - x - x^2 - x^3), {x, 0, n}]; A[] = 0; Do[A[x] = 1 + Sum[t[k] x^k A[x]^k, {k, 2, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=2} A000073(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - x - x^2 - x^3)/(1 - x - x^3)).

A307529 G.f. A(x) satisfies: A(x) = (1 - x^2*A(x)^2)/(1 - x^2*A(x)^2 - x^3*A(x)^3).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 4, 1, 10, 23, 18, 92, 168, 241, 856, 1480, 2904, 8266, 14854, 33496, 83578, 161047, 380488, 884326, 1819714, 4321045, 9730466, 21019404, 49456092, 110408981, 246005440, 572574553, 1281705752, 2906696339, 6711882928, 15128432758, 34625418170
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2019

Keywords

Examples

			G.f.: A(x) = 1 + x^3 + x^5 + 4*x^6 + x^7 + 10*x^8 + 23*x^9 + 18*x^10 + 92*x^11 + 168*x^12 + ...

Crossrefs

Cf. A000931, A049124, A049140, A067955, A307411, A307412, A307413, A307528.

Programs

  • Mathematica
    terms = 37; CoefficientList[1/x InverseSeries[Series[x (1 - x^2 - x^3)/(1 - x^2), {x, 0, terms}], x], x]
terms = 36; A[] = 0; Do[A[x] = (1 - x^2 A[x]^2)/(1 - x^2 A[x]^2 - x^3 A[x]^3) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 37; p[n_] := p[n] = SeriesCoefficient[(1 - x^2)/(1 - x^2 - x^3), {x, 0, n}]; A[] = 1; Do[A[x] = Sum[p[k] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = Sum_{k>=0} A000931(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - x^2 - x^3)/(1 - x^2)).

A367056 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^3*A(x).

Original entry on oeis.org

1, 1, 2, 6, 17, 52, 168, 561, 1922, 6719, 23871, 85938, 312823, 1149421, 4257460, 15880036, 59594517, 224856450, 852491806, 3245959002, 12407332166, 47592364107, 183139542306, 706794663136, 2735053815771, 10609811267757, 41251228784198
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A049140, A071969, A125305, A216490, A367042.

Programs

  • Maple
    A367056 := proc(n)
    add(binomial(n-2*k+1,k) * binomial(2*n-5*k,n-3*k)/(n-2*k+1),k=0..floor(n/3)) ;
end proc:
seq(A367056(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k+1, k)*binomial(2*n-5*k, n-3*k)/(n-2*k+1));

Formula

G.f.: A(x) = 2 / (1-x^3+sqrt((1-x^3)^2-4*x)).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k+1,k) * binomial(2*n-5*k,n-3*k)/(n-2*k+1).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-2*n+7)*a(n-3) +(n-8)*a(n-6)=0. - R. J. Mathar, Dec 04 2023

A367413 Expansion of (1/x) * Series_Reversion( x * (1-x-x^3/(1-x)^2) ).

Original entry on oeis.org

1, 1, 2, 6, 22, 87, 356, 1493, 6398, 27936, 123906, 556734, 2528668, 11590555, 53545932, 249065874, 1165482126, 5482782933, 25914899804, 123009541412, 586121731150, 2802470267460, 13441993044464, 64660400422341, 311861855749484, 1507802756171072, 7306422899878394
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2024

Keywords

Links

Crossrefs

Cf. A378464.
Cf. A049140, A054515.
Cf. A378466, A378467.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(2*n,n-3*k).
From Seiichi Manyama, Nov 27 2024: (Start)
G.f.: exp( Sum_{k>=1} A378464(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x - x^3/(1 - x)^2)^(n+1). (End)

A370797 Expansion of (1/x) * Series_Reversion( x/(x+1/(1-x-x^3)) ).

Original entry on oeis.org

1, 2, 5, 16, 61, 256, 1133, 5191, 24403, 117066, 570835, 2821026, 14097839, 71121660, 361718339, 1852640518, 9547375955, 49469352300, 257564997407, 1346840074300, 7070283106575, 37246786128714, 196849114734855, 1043398553112059, 5545408681615257
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2024

Keywords

Links

Crossrefs

Cf. A218225, A370798.
Cf. A049140.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(x+1/(1-x-x^3)))/x)

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * b(k), where g.f. B(x) = Sum_{k>=0} b(k)*x^k satisfies B(x) = (1/x) * Series_Reversion( x*(1-x-x^3) ).

A052740 A simple context-free grammar in a labeled universe.

Original entry on oeis.org

0, 1, 2, 12, 144, 2400, 50400, 1290240, 39070080, 1365154560, 54047347200, 2391175987200, 116918542540800, 6260970517401600, 364413626331955200, 22906448213096448000, 1546480919558615040000, 111605770820457897984000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Programs

  • Maple
    spec := [S,{C=Prod(B,B),S=Union(B,Z,C),B=Prod(S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Formula

E.g.f.: RootOf(-_Z+_Z^4+_Z^2+x)
D-finite Recurrence: {a(1)=1, a(0)=0, a(2)=2, a(3)=12, (576*n-9216*n^3-2688*n^2+192-6144*n^4)*a(n) +(-6400*n^3-23112-36480*n^2-54608*n)*a(n+1) +(13184*n+17772+896*n^2)*a(n+2) +(-14800-5176*n)*a(n+3) +1147*a(n+4) =0,.
a(n) = n!*A049140(n). - R. J. Mathar, Oct 18 2013

A092413 Coefficient of x^n in solution of x = y + y^2 + y^4 + y^8 + ...

Original entry on oeis.org

1, -1, 2, -6, 20, -70, 256, -970, 3772, -14960, 60280, -246090, 1015700, -4231216, 17767456, -75126078, 319588340, -1366846548, 5873832384, -25350152100, 109828012448, -477486940848, 2082520454864, -9109146150050, 39950535931956
Offset: 1

Views

Author

Ralf Stephan, Mar 22 2004

Keywords

Comments

Reversion of Fredholm-Rueppel sequence (A036987) shifted right.

Crossrefs

Cf. A049140.

Programs

  • Maple
    # Using function CompInv from A357588.
CompInv(25, n -> if 2^ilog2(n) = n then 1 else 0 fi); # Peter Luschny, Oct 05 2022
  • PARI
    serreverse(sum(k=0,8,x^(2^k))+O(x^257))

A371416 Expansion of (1/x) * Series_Reversion( x * (1 - 3*x - x^3) ).

Original entry on oeis.org

1, 3, 18, 136, 1152, 10458, 99472, 978453, 9871686, 101590654, 1062271704, 11253818628, 120535386692, 1303045817184, 14199323523912, 155805565801803, 1720024043803542, 19090440094335912, 212897898182054224, 2384431948345110510, 26808516659219953680
Offset: 0

Views

Author

Seiichi Manyama, Mar 23 2024

Keywords

Links

Crossrefs

Cf. A049140, A120985.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-3*x-x^3))/x)
  • PARI
    a(n) = sum(k=0, n\2, 3^k*binomial(n+k, k)*binomial(4*n+k+2, n-2*k))/(n+1);
  • PARI
    a(n) = sum(k=0, n\3, 3^(n-3*k)*binomial(n+k, k)*binomial(2*n-2*k, n-3*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} 3^k * binomial(n+k,k) * binomial(4*n+k+2,n-2*k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} 3^(n-3*k) * binomial(n+k,k) * binomial(2*n-2*k,n-3*k).
Previous Showing 11-18 of 18 results.

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