A098535 -id:A098535 - cp's OEIS frontend

A098536 Expansion of 1/((1-x)^3 - 9*x^4)^(1/3).

1, 1, 1, 1, 4, 13, 31, 61, 124, 295, 757, 1873, 4402, 10237, 24421, 59701, 146455, 356308, 862810, 2096632, 5127391, 12583513, 30886735, 75775729, 186054142, 457662265, 1127659903, 2781162079, 6862930768, 16945704721, 41876228125, 103573882633, 256359901387, 634900971064

Offset: 0

Paul Barry, Sep 13 2004

Binomial transform of A098535.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A098535, A098538.

Q:=Rationals(); R:=PowerSeriesRing(Q,30); Coefficients(R!(1/((1-x)^3-9*x^4)^(1/3))); // G. C. Greubel, Jan 17 2018

with(FormalPowerSeries): # requires Maple 2022
re:= subs(n=n-1, FindRE(1/((1-x)^3 - 9*x^4)^(1/3), x, a(n)));
# re = Mathar's recurrence
f:= gfun:-rectoproc({re, a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=4}, a(n), remember): map(f, [$0..30]); # Georg Fischer, Oct 23 2022

CoefficientList[Series[1/((1-x)^3-9*x^4)^(1/3),{x,0,40}],x] (* Harvey P. Dale, May 11 2011 *)

x='x+O('x^30); Vec(1/((1-x)^3-9*x^4)^(1/3)) \\ G. C. Greubel, Jan 17 2018

Recurrence: n*a(n) +(-3*n+2)*a(n-1) +(3*n-4)*a(n-2) +(-n+2)*a(n-3) + 3*(-3*n+8)*a(n-4)=0. - R. J. Mathar, Nov 10 2014

a(n) = Sum_{k=0..floor(n/4)} (-9)^k * binomial(-1/3,k) * binomial(n-k,n-4*k). - Seiichi Manyama, Oct 04 2024

A098537 Expansion of (1+x)^(1/3)/(1+x-18*x^4)^(1/3).

Original entry on oeis.org

1, 0, 0, 0, 6, -6, 6, -6, 78, -150, 222, -294, 1374, -3462, 6558, -10662, 30894, -82374, 180222, -339558, 811374, -2082534, 4875774, -10149702, 22872750, -55797126, 133232766, -294821286, 660771438, -1558556070, 3711070590

Offset: 0

Paul Barry, Sep 13 2004

Binomial transform is A098538.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A098535.

Q:=Rationals(); R:=PowerSeriesRing(Q,30); Coefficients(R!((1+x)^(1/3)/(1+x-18*x^4)^(1/3))); // G. C. Greubel, Jan 17 2018

CoefficientList[Series[(1+x)^(1/3)/(1+x-18*x^4)^(1/3), {x, 0, 50}], x] (* G. C. Greubel, Jan 17 2018 *)

a(n):=sum(b(j)*c(n-j,j,0,n); b(n):=if n=0 then 1 else sum(sum((if mod(n-4*k,3)=0 then binomial(k,(4*k-n)/3)*(-1)^((4*k-n)/3)*(18)^((n-k)/3) else 0)*(if k=m then (1/3)^k else m/k*(1/3)^k*sum(binomial(i,k-m-i)*(-1/3)^(k-m-i)*binomial(i+k-1,k-1),i,1,k-m)),k,m,n),m,1,n); c(n):=if n=0 then 1 else (-1)^(n+1)*if n=1 then (1/3)^n else 1/n*(1/3)^n*sum(binomial(k,n-1-k)*(-1/3)^(n-1-k)*binomial(k+n-1,n-1),k,1,n-1);  /* Vladimir Kruchinin, Sep 06 2010 */

x='x+O('x^30); Vec((1+x)^(1/3)/(1+x-18*x^4)^(1/3)) \\ G. C. Greubel, Jan 17 2018

From Vladimir Kruchinin, Sep 06 2010: (Start)
a(n) = Sum(b(j)*c(n-j,j,0,n), where:
b(n) = if n=0 then 1 else Sum(Sum((if mod(n-4*k,3)=0 then binomial(k,(4*k-n)/3)*(-1)^((4*k-n)/3)*(18)^((n-k)/3) else 0)*(if k=m then (1/3)^k else m/k*(1/3)^k*Sum(binomial(i,k-m-i)*(-1/3)^(k-m-i)*binomial(i+k-1,k-1),i,1,k-m)),k,m,n),m,1,n),
c(n)=if n=0 then 1 else (-1)^(n+1)*if n=1 then (1/3)^n else 1/n*(1/3)^n * Sum(binomial(k,n-1-k)*(-1/3)^(n-1-k)*binomial(k+n-1,n-1),k,1,n-1); (End)

cp's OEIS Frontend

A098536 Expansion of 1/((1-x)^3 - 9*x^4)^(1/3).

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