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 1-8 of 8 results.

A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) / (3*k + 1).

Original entry on oeis.org

1, 2, 7, 38, 257, 1935, 15505, 129519, 1115061, 9823160, 88121887, 802227794, 7392428009, 68819554003, 646276497617, 6114880542117, 58237420303109, 557850829527246, 5370956411708779, 51947475492561014, 504492516832543885, 4917564488572565160
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2021

Keywords

Comments

Binomial transform of A002293.

Links

Crossrefs

Cf. A002293, A007317, A104859, A188687, A226974, A346647, A346648, A346649, A346650.

Programs

  • Maple
    A346646 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[2/3,1,4/3],-256/27) ;
    simplify(%) ;
end proc:
seq(A346646(n),n=0..40) ; # R. J. Mathar, Jan 10 2023
  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 21; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {2/3, 1, 4/3}, -256/27], {n, 0, 21}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)/(3*k + 1)); \\ Michel Marcus, Jul 26 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^2 * A(x)^4.
G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 283^(n + 3/2) / (2048 * sqrt(2*Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -2*(2*n-1) *(91*n^2 -91*n +24)*a(n-1) +6*(n-1) *(155*n^2 -310*n +167)*a(n-2) -438*(n-1) *(n-2)*(2*n-3) *a(n-3) +283*(n-1)*(n-2) *(n-3)*a(n-4)=0. - R. J. Mathar, Aug 17 2023

A227035 a(n) = Sum_{k=0..floor(n/4)} binomial(n,4*k)*binomial(5*k,k)/(4*k+1).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 76, 172, 436, 1156, 3006, 7606, 19202, 49466, 130156, 345356, 915196, 2421532, 6427001, 17163581, 46087911, 124133531, 334850208, 904691576, 2449891276, 6651540676, 18100561856, 49344295152, 134719523056, 368350942416, 1008680051756
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 28 2013

Keywords

Comments

Generally, Sum(binomial(n,p*k)*binomial((p+1)*k,k)/(p*k+1), k=0..floor(n/p)) is asymptotic to (p+(p+1)^(1+1/p))^(n+3/2)/(p^(n+1)*(p+1)^(1+3/(2*p))*n^(3/2)*sqrt(2*Pi)).

Links

Crossrefs

Cf. A002294, A007317 (p=1), A049130 (p=2), A226974 (p=3), A226910 (p=5).

Programs

  • Mathematica
    Table[Sum[Binomial[n,4*k]*Binomial[5*k,k]/(4*k+1),{k,0,Floor[n/4]}],{n,0,20}]
  • PARI
    a(n)=sum(k=0,n\4,binomial(n,4*k)*binomial(5*k,k)/(4*k+1)) \\ Charles R Greathouse IV, Jun 28 2013

Formula

Recurrence: -2869*(n-7)*(n-6)*(n-5)*(n-4)*a(n-8) + 2*(n-6)*(n-5)*(n-4)*(5226*n-17267)*a(n-7) - (n-5)*(n-4)*(11582*n^2-55156*n+50139)*a(n-6) - 3*(n-4)*(612*n^3 - 18926*n^2 + 102684*n - 155665)*a(n-5) + 5*(n-4)*(2959*n^3 - 26172*n^2 + 77408*n - 76800)*a(n-4) - 1024*(n-2)*(2*n-5)*(7*n^2-35*n+48)*a(n-3) + 1024*(n-2)*(n-1)*(7*n^2-28*n+30)*a(n-2) - 1024*(n-2)*(n-1)*n*(2*n-3)*a(n-1) + 256*(n-2)*(n-1)*n*(n+1)*a(n) = 0.
a(n) ~ (4+5^(1+1/4))^(n+3/2)/(4^(n+1)*5^(1+3/8)*n^(3/2)*sqrt(2*Pi)).
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^4 * A(x)^5. - Ilya Gutkovskiy, Jul 25 2021
From Peter Bala, Sep 15 2021: (Start)
O.g.f.: A(x) = (1/x)*series reversion ( x*(1 - x^4)/(1 + x*(1 - x^4) )).
The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion ( x*(1 - x^4)/(1 + (m + 1)*x*(1 - x^4)) ). The case m = -1 gives the sequence [1,0,0,0,1,0,0,0,5,0,0,0,35,0,0,0,285,...] - an aerated version of A002294. (End)

A364596 G.f. satisfies A(x) = 1/(1-x) + x^3*(1-x)*A(x)^4.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 15, 36, 82, 191, 471, 1166, 2884, 7267, 18523, 47349, 121821, 315781, 822165, 2148811, 5641035, 14864295, 39287907, 104154066, 276899112, 737984583, 1971375679, 5277570860, 14156881590, 38045460023, 102421374775, 276174537027, 745822179831
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A364595, A364597.
Cf. A215340, A226974.
Cf. A364592, A364594.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-k,2*k) * binomial(4*k,k) / (3*k+1).

A364591 G.f. satisfies A(x) = 1/(1-x) + x^4*A(x)^4.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 40, 85, 197, 457, 1028, 2289, 5193, 12069, 28338, 66445, 155563, 365701, 865815, 2061133, 4919431, 11758741, 28165412, 67657225, 162977081, 393445865, 951438682, 2304494349, 5591221729, 13588455861, 33075115578, 80616857525, 196742749155
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A346073, A364590.
Cf. A226974, A349289, A364588.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-k,3*k) * binomial(4*k,k) / (3*k+1).

A364589 G.f. satisfies A(x) = 1/(1-x) + x^3*A(x)^3.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 31, 67, 146, 331, 760, 1749, 4072, 9583, 22673, 53929, 129055, 310328, 749152, 1815481, 4415313, 10771564, 26352955, 64644926, 158963191, 391767016, 967523138, 2394060433, 5934576763, 14735792889, 36647185192, 91274339014, 227645446307
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A216604, A226974.
Cf. A049130, A199475, A364590.

Programs

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

Formula

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

A369630 Expansion of (1/x) * Series_Reversion( x * (1/(1+x^3) - x) ).

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 255, 960, 3707, 14598, 58395, 236626, 969275, 4007041, 16696822, 70053159, 295691622, 1254772103, 5349978803, 22907982780, 98466168572, 424713570017, 1837717336614, 7974744620620, 34698200181696, 151341512079231, 661590732178716
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2024

Keywords

Links

Crossrefs

Cf. A007863, A199874, A369631.
Cf. A226974.

Programs

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

Formula

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

A370837 Expansion of (1/x) * Series_Reversion( x/(x+1/(1+x^3)) ).

Original entry on oeis.org

1, 1, 1, 0, -3, -9, -15, -6, 57, 231, 501, 474, -1223, -7331, -19655, -27813, 19089, 248541, 819141, 1508316, 417165, -8314449, -34737603, -78646452, -71651147, 251348311, 1461221581, 3984339966, 5586567405, -5424531663, -59608307151, -196443394947
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2024

Keywords

Links

Crossrefs

Cf. A370836, A370838.
Cf. A226974, A370800.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x/(x+1/(1+x^3)))/x)
  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(n, 3*k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n,3*k) * binomial(4*k,k)/(3*k+1).

A364588 G.f. satisfies A(x) = 1/(1-x) + x^2*A(x)^4.

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 170, 613, 2275, 8629, 33301, 130333, 516077, 2063685, 8321892, 33803161, 138181521, 568031297, 2346668400, 9737766513, 40569611691, 169632827345, 711611670532, 2994165070045, 12632782541053, 53433933353885, 226540298098019
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A049130, A090344.
Cf. A226974, A349289, A364591.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n+k,3*k) * binomial(4*k,k) / (3*k+1).
Showing 1-8 of 8 results.

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