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.

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A011245 a(n) = (n+1)*(2*n+1)*(3*n+1)*(4*n+1).

Original entry on oeis.org

1, 120, 945, 3640, 9945, 22176, 43225, 76560, 126225, 196840, 293601, 422280, 589225, 801360, 1066185, 1391776, 1786785, 2260440, 2822545, 3483480, 4254201, 5146240, 6171705, 7343280, 8674225, 10178376, 11870145, 13764520, 15877065, 18223920, 20821801, 23688000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A033593.

Programs

  • GAP
    List([0..30], n-> (n+1)*(2*n+1)*(3*n+1)*(4*n+1) ); # G. C. Greubel, Mar 04 2020
  • Magma
    [&*[s*n+1: s in [1..4]]: n in [0..25]]; // Bruno Berselli, May 23 2011
  • Maple
    seq( mul(j*n+1, j=1..4), n=0..30); # G. C. Greubel, Mar 04 2020
  • Mathematica
    Table[Times@@(Range[4]n+1),{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {1,120,945,3640,9945}, 30] (* Harvey P. Dale, Oct 05 2012 *)
  • PARI
    a(n)=24*n^4+50*n^3+35*n^2+10*n+1 \\ Charles R Greathouse IV, May 23 2011
  • Sage
    [product(j*n+1 for j in (1..4)) for n in (0..30)] # G. C. Greubel, Mar 04 2020

Formula

a(n) = A033593(-n).
G.f.: (1 + 115*x + 355*x^2 + 105*x^3)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(0)=1, a(1)=120, a(2)=945, a(3)=3640, a(4)=9945, a(n) = 5*a(n-1)- 10*a(n-2)+ 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Oct 05 2012
E.g.f.: (1 + 119*x + 353*x^2 + 194*x^3 + 24*x^4)*exp(x). - G. C. Greubel, Mar 04 2020
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=0} 1/a(n) = (4/3 - 3*sqrt(3)/4)*Pi + 12*log(2) - 27*log(3)/4.
Sum_{n>=0} (-1)^n/a(n) = (1 + 4*sqrt(2)/3 - 3*sqrt(3)/2)*Pi - 14*log(2)/3 - 8*sqrt(2)*log(sqrt(2)-1)/3. (End)

A153187 Triangle sequence: T(n, k) = -Product_{j=0..k+1} ((n+1)*j - 1).

Original entry on oeis.org

0, 1, 3, 2, 10, 80, 3, 21, 231, 3465, 4, 36, 504, 9576, 229824, 5, 55, 935, 21505, 623645, 21827575, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200
Offset: 0

Views

Author

Roger L. Bagula, Dec 20 2008

Keywords

Comments

Row sums are: {0, 4, 92, 3720, 239944, 22473720, 2878524564, 483181183072, 102924947692880, 27128289837188700, ...}.

Examples

			Triangle begins as:
  0;
  1,   3;
  2,  10,   80;
  3,  21,  231,  3465;
  4,  36,  504,  9576,  229824;
  5,  55,  935, 21505,  623645,  21827575;
  6,  78, 1560, 42120, 1432080,  58715280,  2818333440;
  7, 105, 2415, 74865, 2919735, 137227545,  7547514975, 475493443425;

Links

Crossrefs

Cf. A001477, A014105, A033593, A033594, A153273.

Programs

  • GAP
    Flat(List([0..10], n-> List([0..n], k-> (-1)*Product([0..k+1], j-> j*(n+1) -1) ))); # G. C. Greubel, Mar 05 2020
  • Magma
    [-(&*[j*(n+1)-1: j in [0..k+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 05 2020
  • Maple
    seq(seq(-mul(j*(n+1)-1, j = 0..k+1), k = 0..n), n = 0..10); # G. C. Greubel, Mar 05 2020
  • Mathematica
    T[n_, m_] = -Product[(n+1)*j -1, {j,0,m+1}]; Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten
Table[-(n+1)^(k+2)*Pochhammer[-1/(n+1), k+2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 05 2020 *)
  • PARI
    T(n,k) = (-1)*prod(j=0, k+1, j*(n+1)-1);
for(j=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 05 2020
  • Sage
    [[-(n+1)^(k+2)*rising_factorial(-1/(n+1), k+2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 05 2020

Formula

T(n, k) = -Product_{j=0..k+1} (j*(n+1) - 1).
T(n, k) = -(n+1)^(k+2) * Pochhammer(-1/(n+1), k+2).

Extensions

Edited by G. C. Greubel, Mar 05 2020
Showing 1-2 of 2 results.

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

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