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 1-3 of 3 results.

A259461 From higher-order arithmetic progressions: Erroneous version of A378234.

Original entry on oeis.org

40, 5000, 472500, 43218000, 4148928000, 432170216000, 49514201000000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2015

Keywords

Comments

Only the first 5 terms in the publication are correct. The corrected sequence would start 40, 5000, 472500, 43218000, 4148928000, 432081216000, 49509306000000, 6275893932000000, 881135508052800000,... - R. J. Mathar, Jul 14 2015
The corrected version is A378234, - Georg Fischer, Dec 17 2024

Crossrefs

Cf. A378234.

A259459 From higher-order arithmetic progressions.

Original entry on oeis.org

1, 18, 360, 9000, 283500, 11113200, 533433600, 30862944000, 2121827400000, 171160743600000, 16020645600960000, 1722947613266880000, 211061082625192800000, 29223842209642080000000, 4542220046298654720000000, 787620956028186728448000000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2015

Keywords

Comments

The expression "2 over n!" in the article is A006472(n+1). It is used in A259459 - A378234 (C_1 - C_3) on page 13. - Georg Fischer, Dec 06 2024

Crossrefs

Programs

  • Maple
    rV := proc(n,a,d)
            n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
    end proc:
    A259459 := proc(n)
            mul(rV(i,a,d),i=1..n+1) ;
            coeftayl(%,d=0,1) ;
            coeftayl(%,a=0,n) ;
    end proc:
    seq(A259459(n),n=1..15) ; # R. J. Mathar, Jul 14 2015
  • Mathematica
    rV[n_, a_, d_] := n(n+1)/2*a + (n-1)n(n+1)/6*d;
    A259459[n_] :=
       Product[rV[i, a, d], {i, 1, n+2}] //
       SeriesCoefficient[#, {d, 0, 1}]& //
       SeriesCoefficient[#, {a, 0, n+1}]&;
    Table[A259459[n], {n, 0, 14}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

Formula

-2*n*a(n) +(n+3)*(n+2)^2*a(n-1)=0. - R. J. Mathar, Jul 15 2015
Conjectured g.f.: 3F0(4,3,3;;x/2). - R. J. Mathar, Aug 09 2015
a(n) = (n+3)!*(n+2)!/2^(n+2)*(n+1)*(n+2)/6. - Georg Fischer, Dec 06 2024

A259460 From higher-order arithmetic progressions.

Original entry on oeis.org

4, 220, 10500, 535500, 30870000, 2044828800, 156029328000, 13673998800000, 1369285948800000, 155756276676000000, 20005336176265440000, 2884501462544301600000, 464334381775424160000000, 83021688624014300160000000, 16408769917253890176000000000, 3569104362241728159962112000000, 850861011640079911341911040000000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2015

Keywords

Comments

The expression "2 over n!" in the article is A006472(n+1). It is used in C_1, C_2, C_3 (A259459, A259460, A378234) on page 13. A_2 is A000914. - Georg Fischer, Dec 06 2024

Crossrefs

Programs

  • Maple
    rV := proc(n,a,d)
        n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
    end proc:
    A259460 := proc(n)
        mul(rV(i,a,d),i=1..n+2) ;
        coeftayl(%,d=0,2) ;
        coeftayl(%,a=0,n) ;
    end proc:
    seq(A259460(n),n=1..18) ; # R. J. Mathar, Jul 14 2015
  • Mathematica
    rV[n_, a_, d_] := n (n + 1)/2*a + (n - 1) n (n + 1)/6*d;
    A259460[n_] :=
       Product[rV[i, a, d], {i, 1, n + 3}] //
       SeriesCoefficient[#, {d, 0, 2}] & //
       SeriesCoefficient[#, {a, 0, n + 1}] & ;
    Table[A259460[n], {n, 0, 16}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

Formula

From Georg Fischer, Dec 06 2024: (Start)
a(n) = (n+4)!*(n+3)!/2^(n+3)/9 * (n+2)*(n+1)*(n+3)*(3*n+8)/24.
D-finite with recurrence: 2*n*(3*n+5)*a(n) - (n+3)^2*(n+4)*(3*n+8)*a(n-1) = 0. (End)

Extensions

Typos in data corrected by Jean-François Alcover, Apr 27 2023
Showing 1-3 of 3 results.