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 1-4 of 4 results.

A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).

Original entry on oeis.org

1, 4, 22, 146, 1079, 8525, 70468, 601816, 5268241, 47019566, 426250277, 3914020148, 36328457669, 340278596273, 3212416054283, 30534649412247, 291981031204917, 2806832429353512, 27109863184695640, 262951127248539898, 2560229132085602215
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2024

Keywords

Crossrefs

Programs

  • Maple
    A369617 := proc(n)
        add(binomial(n+1,k) * binomial(4*n-4*k+2,n-k),k=0..n) ;
        %/(n+1) ;
    end proc;
    seq(A369617(n),n=0..70) ; # R. J. Mathar, Jan 28 2024
  • 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, binomial(n+1, k)*binomial(4*n-4*k+2, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(4*n-4*k+2,n-k).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-91*n^3 -32*n^2 +n+2)*a(n-1) +2*(n-1)*(465*n^2 -337*n+86)*a(n-2) -4*(n-1)*(n-2) *(219*n-187)*a(n-3) +283*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 28 2024

A379186 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^3) * (1 - x*A(x))^2).

Original entry on oeis.org

1, 3, 21, 202, 2270, 27903, 363412, 4927840, 68834941, 983680783, 14312988289, 211329419670, 3158263216267, 47682769300288, 726188701482730, 11142842570134264, 172101193009427174, 2673445730846829604, 41742159037922167264, 654721526817143247304, 10311337739352708700427
Offset: 0

Views

Author

Seiichi Manyama, Dec 17 2024

Keywords

Crossrefs

Programs

  • Mathematica
    terms = 21; A[] = 0; Do[A[x] = 1/((1-x*A[x]^3)*(1 -x*A[x])^2) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jun 14 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(3*n+3*k+1, n-k)/(n+3*k+1));

Formula

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

A370844 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^4 + x) ).

Original entry on oeis.org

1, 5, 35, 295, 2760, 27556, 287564, 3098780, 34216020, 385106280, 4401850866, 50957904938, 596231618166, 7039674475190, 83767631913840, 1003564049999916, 12094813260406732, 146534778450346908, 1783695235540931924, 21803615393276536720, 267537602528379374851
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^4+x))/x)
    
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial(5*k+3, k)/(k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A118971(k).
a(n) = hypergeom([4/5, 6/5, 7/5, 8/5, -n], [5/4, 3/2, 7/4, 2], -3125/256). - Stefano Spezia, Mar 03 2024

A379185 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x))^2).

Original entry on oeis.org

1, 3, 18, 139, 1222, 11618, 116372, 1209779, 12930966, 141225530, 1569136588, 17680779230, 201562070356, 2320574126216, 26942875644408, 315109464849603, 3708926665685286, 43901133108206978, 522240410257549260, 6240258006163094026, 74864641626913850964
Offset: 0

Views

Author

Seiichi Manyama, Dec 17 2024

Keywords

Crossrefs

Programs

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

Formula

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