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.

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

Links

Crossrefs

Cf. A000914, A006472, A259459, A378234.

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

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

Original entry on oeis.org

40, 5000, 472500, 43218000, 4148928000, 432081216000, 49509306000000, 6275893932000000, 881135508052800000, 136878615942868800000, 23474682634201999200000, 4432282735129048800000000, 918537831584839065600000000, 208281986149676045967360000000, 51516317681413623440962560000000
Offset: 0

Views

Author

Georg Fischer, Dec 16 2024

Keywords

Comments

Only the first 5 terms of A259461 are correct. - R. J. Mathar, Jul 14 2015
"2 over n!" on page 13 in the Dienger article is A006472; A_3 is A001303.

Links

Crossrefs

Cf. A001303, A006472, A259459, A259460, A259461.

Programs

  • Maple
    rV := proc(n,a,d)
    n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259461 := proc(n)
    mul(rV(i,a,d),i=1..n+3) ;
    coeftayl(%,d=0,3) ;
    coeftayl(%,a=0,n) ;
end proc:
seq(A259461(n),n=1..5) ; # R. J. Mathar, Jul 14 2015

Formula

D-finite with recurrence: -2*n*(n+2)*a(n) + (n+4)^3*(n+5)*a(n-1) = 0.
a(n) = (n+5)!*(n+4)!^3 / (1296*2^(n+4)*n!^2*(n+2)*(n+1)).

A139769 T(n,k) = [x^k] Product_{m=1..n} d/dx Sum_{i=1..m} x^i; triangle read by rows, n >= 0, 0 <= k <= A161680(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 7, 6, 1, 6, 18, 36, 49, 46, 24, 1, 8, 33, 94, 204, 354, 497, 562, 501, 326, 120, 1, 10, 52, 188, 528, 1222, 2406, 4102, 6116, 7996, 9132, 9014, 7541, 5116, 2556, 720, 1, 12, 75, 326, 1105, 3106, 7513, 16014, 30558, 52752, 82938, 119230, 156983
Offset: 0

Views

Author

Roger L. Bagula, Jun 13 2008

Keywords

Comments

Row sums are A006472(n+1).
T(n, binomial(n,2)-k) is the number of rank-k intervals in the middle order on permutations. (See Bouvel et al. reference.) - Bridget Tenner, May 24 2024

Examples

			Triangle T(n,k) begins:
  1;
  1;
  1, 2;
  1, 4,  7,  6;
  1, 6, 18, 36,  49,  46,  24;
  1, 8, 33, 94, 204, 354, 497, 562, 501, 326, 120;
  ...

Links

Crossrefs

Cf. A000142, A008302 (Mahonian numbers), A006472, A010551, A161680, A259459.

Programs

  • Mathematica
    a := Table[CoefficientList[Product[Sum[D[x^i, x], {i, 1, m}], {m, 1, n}], x], {n, 0, 7}]; Flatten[a]

Formula

From Alois P. Heinz, May 24 2024: (Start)
|Sum_{k=0..binomial(n,2)} (-1)^k T(n,k)| = A010551(n).
Sum_{k=0..binomial(n,2)} (binomial(n,2)-k)*T(n,k) = A259459(n-2) for n>=2. (End)

Extensions

Edited by Alois P. Heinz, May 24 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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