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.

A033593 a(n) = (n-1)*(2*n-1)*(3*n-1)*(4*n-1).

Original entry on oeis.org

1, 0, 105, 880, 3465, 9576, 21505, 42120, 74865, 123760, 193401, 288960, 416185, 581400, 791505, 1053976, 1376865, 1768800, 2238985, 2797200, 3453801, 4219720, 5106465, 6126120, 7291345, 8615376, 10112025, 11795680, 13681305, 15784440, 18121201, 20708280, 23562945
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The sequence of n such that n is prime and (2*n+1) is prime is the sequence of Sophie Germain primes A005384; the subsequence of those for which in addition (3*n+2) is prime is A067256; and the subsequence of those for which in addition (4*n+3) is prime is A067257. - Jonathan Vos Post, Dec 15 2004

Links

Crossrefs

a(n) = A011245(-n).
Cf. A005384, A033594, A067256, A067257.

Programs

  • Magma
    [ 24*n^4-50*n^3+35*n^2-10*n+1: n in [0..40]]; // Vincenzo Librandi, Jan 30 2011
  • Magma
    [&*[s*n-1: s in [1..4]]: n in [0..40]]; // Bruno Berselli, May 23 2011
  • Maple
    1, seq( n^4*pochhammer((n-1)/n, 4), n=1..40); # G. C. Greubel, Mar 05 2020
  • Mathematica
    Table[1-10 n+35 n^2-50 n^3+24 n^4,{n,0,40}] (* or *) LinearRecurrence[{5,-10, 10,-5,1}, {1,0,105,880,3465}, 40]  (* Harvey P. Dale, Jan 29 2011 & Apr 26 2011 *)
  • PARI
    a(n)=24*n^4-50*n^3+35*n^2-10*n+1 \\ Charles R Greathouse IV, May 23 2011
  • Sage
    [1]+[n^4*rising_factorial((n-1)/n, 4) for n in (1..40)] # G. C. Greubel, Mar 05 2020

Formula

G.f.: (1 -5*x +115*x^2 +345*x^3 +120*x^4)/(1-x)^5. - R. J. Mathar, Jan 30 2011
From G. C. Greubel, Mar 05 2020: (Start)
a(n) = n^4* Pochhammer((n-1)/n, 4).
E.g.f.: (1 - x + 53*x^2 + 94*x^3 + 24*x^4)*exp(x). (End)
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=2} 1/a(n) = 29/36 + (4/3 - 3*sqrt(3)/4)*Pi - 12*log(2) + 27*log(3)/4.
Sum_{n>=2} (-1)^n/a(n) = (1 + 4*sqrt(2)/3 - 3*sqrt(3)/2)*Pi + 14*log(2)/3 - 4*sqrt(2)*log(2)/3 + 8*sqrt(2)*log(2-sqrt(2))/3 - 29/36. (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

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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

A153273 Triangle read by rows: T(n,k) = Product_{i=0..k-2} (i*n + n - 1).

Original entry on oeis.org

1, 2, 10, 3, 21, 231, 4, 36, 504, 9576, 5, 55, 935, 21505, 623645, 6, 78, 1560, 42120, 1432080, 58715280, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 9, 171, 4959, 193401, 9476649, 559122291, 38579438079, 3047775608241, 271252029133449
Offset: 2

Views

Author

Roger L. Bagula, Dec 22 2008

Keywords

Comments

Row sums are {1, 12, 255, 10120, 646145, 60191124, 7687739647, 1288641721680, 274338952977249, 72299818200530140, ...}.
A153187 without its diagonal. - R. J. Mathar, Sep 04 2016

Examples

			Triangle begins as:
  1;
  2,  10;
  3,  21,  231;
  4,  36,  504,   9576;
  5,  55,  935,  21505,  623645;
  6,  78, 1560,  42120, 1432080,  58715280;
  7, 105, 2415,  74865, 2919735, 137227545,  7547514975;
  8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640;

Links

Crossrefs

Cf. A000027, A014105, A033594.

Programs

  • GAP
    Flat(List([2..12], n-> List([2..n], k-> Product([0..k-2], j-> (j+1)*n-1) ))); # G. C. Greubel, Mar 05 2020
  • Magma
    [(&*[j*n+n-1: j in [0..k-2]]): k in [2..n], n in [2..12]]; // G. C. Greubel, Mar 05 2020
  • Maple
    A153273 := proc(n,m)
    local i;
    mul( n-1+i*n, i=0..m-2) ;
end proc:
seq(seq( A153273(n,m), m=2..n), n=2..12) ; # R. J. Mathar, Sep 04 2016
  • Mathematica
    Table[n^(k-1)*Pochhammer[(n-1)/n, k-1], {n,2,12}, {k,2,n}]//Flatten (* modified by G. C. Greubel, Mar 05 2020 *)
  • PARI
    T(n,k) = prod(j=0, k-2, j*n+n-1);
for(n=2,12, for(k=2,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 05 2020
  • Sage
    [[n^(k-1)*rising_factorial((n-1)/n, k-1) for k in (2..n)] for n in (2..12)] # G. C. Greubel, Mar 05 2020

Extensions

Edited by G. C. Greubel, Mar 05 2020
Showing 1-3 of 3 results.

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

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