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 1-7 of 7 results.

A218622 a(n) = A183161(n) (mod 4), n>=0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 0, 0, 2, 2, 2, 0, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2012

Keywords

Comments

Conjecture: a(n) never equals 3.
A183161(n) is defined by the convolution:
Sum_{k=0..n} A183161(n-k)*A183161(k) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k).
The g.f. F(x) of A183161 satisfies: F(x) = 1/sqrt(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764.

Examples

			Formatting the terms into groups of 8 reveals complex binary patterns:
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 0,0,0,0,0,0,0,0,
2,2,2,0,2,2,0,0, 2,2,2,0,0,0,0,0, 2,2,2,0,2,2,0,0, 2,2,2,0,2,2,2,2,
1,1,1,2,1,1,2,2, 1,1,1,0,2,2,2,2, 1,1,1,2,1,1,0,0, 2,2,2,0,2,2,2,2, ...

Links

Crossrefs

Cf. A183161.

Programs

  • PARI
    {a(n)=local(A2=sum(m=0, n, sum(k=0, m, binomial(m+k, m-k)*binomial(2*m-k, k))*x^m+x*O(x^n))); polcoeff(A2^(1/2), n)%4}
  • PARI
    {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/sqrt(1-2*x*G^2-3*x^2*G^4), n)%4}
  • PARI
    /* Using Central Trinomial Coefficients A002426: */
{A002426(n)=sum(k=0, n\2, binomial(n, 2*k)*binomial(2*k, k))}
{a(n)=if(n==0, 1, sum(k=0, n, A002426(k)*binomial(3*n-k, n-k)*2*k/(3*n-k)))%4}
/* Format Print of a(n) into 4 columns of 8 terms each: */
for(n=0,1024,if(n>0,if(n%32==0,print(""),if(n%8==0,print1(" "))));print1(a(n),","))

A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k).

Original entry on oeis.org

1, 2, 11, 62, 367, 2232, 13820, 86662, 548591, 3498146, 22436251, 144583496, 935394436, 6071718512, 39523955552, 257913792342, 1686627623151, 11050540084902, 72522925038257, 476669316338542, 3137209052543927, 20672732229560032, 136374124374593072, 900541325129687272
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2010

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 +...
A(x)^(1/2) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...+ A183161(n)*x^n +...
Given triangle A085478(n,k) = C(n+k,n-k), which begins:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  5,  1;
  1, 10, 15,  7, 1;
  1, 15, 35, 28, 9, 1; ...
ILLUSTRATE formula a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k):
a(2) = 11 = 1*1 + 3*3 + 1*1;
a(3) = 62 = 1*1 + 6*5 + 5*6 + 1*1;
a(4) = 367 = 1*1 + 10*7 + 15*15 + 7*10 + 1*1;
a(5) = 2232 = 1*1 + 15*9 + 35*28 + 28*35 + 9*15 + 1*1; ...

Links

Crossrefs

Cf. A001764, A085478, A183161, A199033, A218622.
Cf. A026641, A147855, A371753.
Cf. A005809, A006256, A160906, A226751.
Cf. A171822.

Programs

  • Magma
    [(&+[Binomial(n+k, 2*k)*Binomial(2*n-k, k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 22 2021
  • Mathematica
    Table[Sum[Binomial[n+k,n-k]Binomial[2n-k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)
Table[HypergeometricPFQ[{-n, -n, 1/2 -n, n+1}, {1/2, 1, -2*n}, 1], {n, 0, 25}] (* G. C. Greubel, Feb 22 2021 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(2*n-k,k))}
  • PARI
    {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1-2*x*G^2-3*x^2*G^4), n)} \\ Paul D. Hanna, Nov 03 2012
  • PARI
    {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1+3*x*G-5*x*G^2), n)} \\ Paul D. Hanna, Jun 16 2013
for(n=0, 30, print1(a(n), ", "))
  • Sage
    a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],2)
[simplify(a(n)) for n in range(26)] # Peter Luschny, May 19 2015

Formula

a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k).
Self-convolution of A183161 (an integer sequence):
a(n) = Sum_{k=0..n} A183161(k)*A183161(n-k).
a(n) = Sum_{k=0..n} binomial(2*n+k,k) * cos((n+k)*Pi). - Arkadiusz Wesolowski, Apr 02 2012
Recurrence: 320*n*(2*n-1)*a(n) = 8*(346*n^2 + 79*n - 327)*a(n-1) + 6*(1688*n^2-6241*n+5981)*a(n-2) + 261*(3*n-7)*(3*n-5)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3^(3*n+3/2)/(2^(2*n+3)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
...
G.f.: A(x) = 1/(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012
G.f.: A(x) = 1 + x*d/dx { log( G(x)^5/(1+x*G(x)^2) )/2 }, where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 04 2012
G.f.: A(x) = 1/(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Jun 16 2013
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],2). - Peter Luschny, May 19 2015
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n)). - Ilya Gutkovskiy, Oct 25 2017
From G. C. Greubel, Feb 22 2021: (Start)
a(n) = Sum_{k=0..n} A171822(n, k).
a(n) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1). (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Apr 05 2024
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
G.f.: g^2/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)
G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025

A219197 Self-convolution equals A199033.

Original entry on oeis.org

1, 2, 9, 46, 253, 1452, 8570, 51594, 315225, 1948010, 12147881, 76316508, 482392198, 3064987460, 19560379470, 125309993974, 805458510441, 5192500350906, 33561539356277, 217429403317006, 1411572472199649, 9181398851046632, 59821825063376124, 390382132833183204
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2012

Keywords

Comments

Conjecture: a(n) is never congruent to 3 modulo 4.

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 46*x^3 + 253*x^4 + 1452*x^5 +...
where A(x)^2 = 1 + 4*x + 22*x^2 + 128*x^3 + 771*x^4 + 4744*x^5 +...+ A199033(n)*x^n +...
Also, the g.f. A(x) satisfies: A(x) = G(x) * F(x*G(x)^2) where
F(x) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 141*x^6 +...+ A002426(n)*x^n +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...+ A001764(n)*x^n +...

Links

Crossrefs

Cf. A199033, A183161, A002426.

Programs

  • Mathematica
    A002426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}]; Table[Sum[A002426[k]*Binomial[3*n - k + 1, n - k]*(2*k + 1)/(3*n - k + 1), {k, 0, n}], {n, 0, 50} ] (* G. C. Greubel, Mar 06 2017 *)
  • PARI
    {a(n)=local(A2=sum(m=0, n, sum(k=0, m, binomial(m+k+1, m-k)*binomial(2*m-k+1, k))*x^m+x*O(x^n))); polcoeff(A2^(1/2), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(G/sqrt(1-2*x*G^2-3*x^2*G^4), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {A002426(n)=sum(k=0, n\2, binomial(n, 2*k)*binomial(2*k, k))}
{a(n)=if(n==0, 1, sum(k=0, n, A002426(k)*binomial(3*n-k+1, n-k)*(2*k+1)/(3*n-k+1)))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {A097893(n)=sum(m=0, n, sum(k=0, m\2, binomial(m, 2*k)*binomial(2*k, k)))}
{a(n)=if(n==0, 1, sum(k=0, n, A097893(k)*binomial(3*n-k, n-k)*2*k/(3*n-k)))}
for(n=0, 30, print1(a(n), ", "))

Formula

Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(n+k+1,n-k)*C(2*n-k+1,k) = A199033(n).
G.f.: A(x) = G(x) / sqrt(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764.
G.f.: A(x) = Sum_{n>=0} A002426(n) * x^n * G(x)^(2*n+1), where A002426 are the central trinomial coefficients and G(x) = 1 + x*G(x)^3 = g.f. of A001764.
a(n) = Sum_{k=0..n} A002426(k) * C(3*n-k+1,n-k) * (2*k+1)/(3*n-k+1) for n>0, where A002426 are the central trinomial coefficients.
From Vaclav Kotesovec, Oct 05 2020: (Start)
Recurrence: 32*(n-1)*n*(2*n + 1)*(49*n^2 - 210*n + 222)*a(n) = 4*(n-1)*(10388*n^4 - 55104*n^3 + 96925*n^2 - 64446*n + 15006)*a(n-1) - 6*(22050*n^5 - 173439*n^4 + 536588*n^3 - 814340*n^2 + 604331*n - 174702)*a(n-2) - 81*(n-2)*(3*n - 7)*(3*n - 5)*(49*n^2 - 112*n + 61)*a(n-3).
a(n) ~ 3^(3*n + 7/4) / (Gamma(1/4) * n^(3/4) * 2^(2*n + 5/2)). (End)

A184554 Self-convolution equals A184553.

Original entry on oeis.org

1, 3, 35, 474, 6891, 104360, 1623050, 25718472, 413215707, 6710439939, 109904635992, 1812533851286, 30064278051066, 501094410251724, 8386624585529736, 140867399832201392, 2373517896651329211
Offset: 0

Views

Author

Paul D. Hanna, Jan 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 35*x^2 + 474*x^3 + 6891*x^4 + 104360*x^5 +...
A(x)^2 = 1 + 6*x + 79*x^2 + 1158*x^3 + 17851*x^4 + 283246*x^5 + 4579306*x^6 +...+ A184553(n)*x^n +...

Links

Crossrefs

Cf. A184553, A183161.

Programs

  • PARI
    {a(n)=local(A2=sum(m=0, n, sum(k=0, m, binomial(3*m+k, m-k)*binomial(4*m-k,k))*x^m+x*O(x^n))); polcoeff(A2^(1/2), n)}

Formula

Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(3n+k,n-k)*C(4n-k,k).
From Vaclav Kotesovec, Oct 05 2020: (Start)
Recurrence: 214990848*(n-1)*n*(2*n - 1)*(3*n - 2)*(3*n - 1)*(6*n - 5)*(6*n - 1)*(186273819397795248*n^10 - 3808534256911136592*n^9 + 34796629832777934044*n^8 - 187043184670288993620*n^7 + 654924444499586105253*n^6 - 1560497773606771079631*n^5 + 2561855600168977896561*n^4 - 2860703663001433319865*n^3 + 2078954085287299115314*n^2 - 887684422175942220312*n + 169072062403455034560)*a(n) = 4608*(n-1)*(198724237800527802240275328*n^16 - 4858079103240534545068477440*n^15 + 54642880050599236064003042400*n^14 - 374971611530329495568194400064*n^13 + 1755012126559348835693811861336*n^12 - 5932874750520572573109360538464*n^11 + 14963200004531199654932033618980*n^10 - 28673115580838375915179786787512*n^9 + 42110417881849893372861579821049*n^8 - 47452926131517602493132103229892*n^7 + 40787440281095755641655153245870*n^6 - 26377403019510743431336185730824*n^5 + 12533524812443184688591954943537*n^4 - 4209478808341707391000183637804*n^3 + 936469297986509962567040719500*n^2 - 122249305263206439707648569200*n + 6971356419682529260674288000)*a(n-1) - 56*(144127472482539322780646079360*n^17 - 3883898817531102082800910713408*n^16 + 48508159847016350153460641621824*n^15 - 372595836943650102686279598364576*n^14 + 1969341905770556875691823842890528*n^13 - 7592461765467638827554889057000528*n^12 + 22081067675956591873212706692285658*n^11 - 49409399253505762430059464298609003*n^10 + 85975527766708349125221818435201054*n^9 - 116774106223860017685029870069674610*n^8 + 123519332779512059055811665446677502*n^7 - 100900018919730898675966601457772931*n^6 + 62669067428714316682185812483083058*n^5 - 28859522902379638250482670957736704*n^4 + 9468958420673660795040733200084216*n^3 - 2072765338981605418905542909841840*n^2 + 268050646552599842334757631726400*n - 15244779045642670870731418176000)*a(n-2) - 5764801*(n-2)*(7*n - 20)*(7*n - 19)*(7*n - 18)*(7*n - 17)*(7*n - 16)*(7*n - 15)*(186273819397795248*n^10 - 1945796062933184112*n^9 + 8902143393478490876*n^8 - 23424520929131008820*n^7 + 39128411618346831497*n^6 - 43181027932317815765*n^5 + 31728210165988475069*n^4 - 15234320478860939075*n^3 + 4539474588760473630*n^2 - 750365114092889508*n + 51518300147130960)*a(n-3).
a(n) ~ 7^(7*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(6*n + 2) * 3^(6*n + 1/4)).
(End)

A385719 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.

Original entry on oeis.org

1, 4, 38, 428, 5204, 66104, 863840, 11515308, 155779966, 2131436392, 29426804398, 409254436452, 5726378247412, 80535621269208, 1137609359823936, 16130112288879248, 229462608491483364, 3273749607191060480, 46826932120849617128, 671341041479214814160, 9644654058165119642624
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2025

Keywords

Links

Crossrefs

Cf. A183161, A208977, A387084, A387086.
Cf. A002295, A004355, A385497.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] - 1)/3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)

Formula

Sum_{k=0..n} a(k) * a(n-k) = A385497(n).
G.f.: 1/sqrt(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g/sqrt((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
a(n) ~ 2^(6*n - 1/2) * 3^(6*n + 3/4) / (Gamma(1/4) * n^(3/4) * 5^(5*n + 1/4)) * (1 + 7*Gamma(1/4)^2/(48*Pi*sqrt(30*n))). - Vaclav Kotesovec, Aug 20 2025

A387084 Expansion of B(x)/sqrt(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449.

Original entry on oeis.org

1, 3, 23, 211, 2095, 21752, 232439, 2534182, 28041295, 313833025, 3544160216, 40318629754, 461455158383, 5308453068900, 61333295856750, 711305543582150, 8276351877367663, 96576953297406377, 1129842469637643485, 13248082583624602575, 155660344852055352760
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Crossrefs

Cf. A183161, A208977, A387086.
Cf. A001449, A002294, A079589.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Sum[Binomial[5*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 4*(Sum[Binomial[5*n, n]*x^n, {n, 0, nmax}] - 1)/5], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)

Formula

Sum_{k=0..n} a(k) * a(n-k) = A079589(n).
G.f.: 1/sqrt(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g/sqrt(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.
Conjecture D-finite with recurrence 3902464*n*(8*n-5) *(8*n-3)*(8*n-1) *(8*n+1)*a(n) +80*(-12565760000*n^5 +68448000000*n^4 -163457516000*n^3 +200475354000*n^2 -122843089511*n +29804717943)*a(n-1) +125000*(134055000*n^5 -1109795000*n^4 +3726971625*n^3 -6307124125*n^2 +5325821766*n -1769460798)*a(n-2) +48828125*(-1556875*n^5 +15845625*n^4 -60659875*n^3 +103818375*n^2 -67764178*n +1391424)*a(n-3) -152587890625 *(5*n-16)*(n-3) *(5*n-19)*(5*n-18) *(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 19 2025
a(n) ~ 5^(5*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(8*n + 7/4)). - Vaclav Kotesovec, Aug 20 2025

A387086 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.

Original entry on oeis.org

1, 0, 2, 4, 16, 52, 188, 672, 2458, 9052, 33648, 125864, 473500, 1789632, 6791528, 25863568, 98796096, 378411332, 1452886052, 5590262688, 21551271916, 83228809640, 321933018272, 1247062996304, 4837152438556, 18785529571200, 73037938668632, 284268423472432
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Crossrefs

Cf. A183161, A208977, A387084.
Cf. A000108, A000984, A387085.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] - 1)], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(4*x-1+2*sqrt(1-4*x)))

Formula

Sum_{k=0..n} a(k) * a(n-k) = A387085(n).
G.f.: 1/sqrt( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/sqrt(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g/sqrt((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
a(n) ~ 2^(2*n - 1/2) / (Gamma(1/4) * n^(3/4)) * (1 - Gamma(1/4)^2/(16*Pi*sqrt(2*n))). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*(n-1)*a(n) -2*(n-1)*(10*n-17)*a(n-1) +4*(4*n^2-24*n+29)*a(n-2) +32*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025
Showing 1-7 of 7 results.

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