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.

A125650 Numerator of n(n+3)/(4(n+1)(n+2)) = sum(k=1..n, 1/(k(k+1)(k+2)) ).

Original entry on oeis.org

0, 1, 5, 9, 7, 5, 27, 35, 11, 27, 65, 77, 45, 26, 119, 135, 38, 85, 189, 209, 115, 63, 275, 299, 81, 175, 377, 405, 217, 116, 495, 527, 140, 297, 629, 665, 351, 185, 779, 819, 215, 451, 945, 989, 517, 270, 1127, 1175, 306, 637, 1325, 1377, 715, 371, 1539, 1595, 413, 855
Offset: 0

Views

Author

Alexander Adamchuk, Nov 29 2006

Keywords

Comments

3^2 divides a(3k). p divides a(p) for an odd prime p. p divides a(p-3) for prime p>3. p^k divides a(p^k) for an odd prime p. a(n) = m^2 is a perfect square for n = {1,3,24,147,864,5043,29400,171363,...} = A125651(n). Corresponding numbers m such that m^2 = a[ A125651(n) ] are listed in A125652(n) = {1,3,9,105,306,3567,10395,121173,...}.

Examples

			The rationals n(n+3)/(4(n+1)(n+2)) = a(n)/A230328(n) begin:
0, 1/6, 5/24, 9/40, 7/30, 5/21, 27/112, 35/144, 11/45, 27/110, 65/264, 77/312, 45/182, 26/105, 119/480, ... - _Wolfdieter Lang_, Mar 08 2018

References

  • L. B.W. Jolley, Summation of Series, Second revised ed., Dover, 1961, p.38, (201).

Links

Crossrefs

Cf. A125651, A125652. A160050, A230328 (denominators).

Programs

  • Magma
    [Numerator(n*(n+3)/(4*(n+1)*(n+2))): n in [0..60]]; // Vincenzo Librandi, May 21 2012
  • Mathematica
    Table[Numerator[n(n+3)/(4(n+1)(n+2))],{n,0,100}]
  • PARI
    a(n)=n*(n+3)/2^min(3,valuation(n*(n+3),2)); \\ Max Alekseyev, Jan 11 2007

Formula

a(n) = Sum_{k=1..n} 1/(k(k+1)(k+2)).
a(n) = n*(n+3)/2^min(3,valuation(n*(n+3),2)). a(n)=n*(n+3)/4 for n=1 or 4 (mod 8); a(n)=n*(n+3)/8 for n=0 or 5 (mod 8); a(n) = n*(n+3)/2 for n=2, 3, 6, or 7 (mod 8). - Max Alekseyev, Jan 11 2007
a(n) = A106609(n)*A106609(n+3). - Paul Curtz, Jan 13 2011
G.f.: x*(x^19 -2*x^18 +3*x^17 -5*x^16 +3*x^15 -6*x^14 +7*x^13 -11*x^12 -12*x^11 +24*x^10 -36*x^9 +24*x^8 -38*x^7 +28*x^6 -18*x^5 -3*x^4 -2*x -1) / ((x-1)^3*(x^2+1)^3*(x^4+1)^3). - Colin Barker, Feb 21 2013
G.f. for rationals r(n) = a(n)/A230328(n): (1/4)*(1 - hypergeometric([1, 2], [3], -x/(1-x)))/(1-x) = (- 2*x + 3*x^2 + 2*(2*x - (1 + x^2))*log(1-x))/(4*(1-x)*x^2). For the r(n) formula see Jolley's general remark (201) on p.38. Thanks to Gary Detlefs for pointing to this remark. - Wolfdieter Lang, Mar 08 2018

A230339 Numerator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Original entry on oeis.org

0, 1, 1, 19, 17, 55, 83, 119, 82, 73, 95, 121, 227, 559, 679, 815, 484, 1139, 443, 171, 295, 2023, 2299, 2599, 1462, 3275, 3653, 451, 749, 551, 5455, 5983, 3272, 7139, 7769, 8435, 1523, 3293, 3553, 11479, 6170, 13243, 14189, 15179, 8107, 5765
Offset: 0

Views

Author

Jean-François Alcover, Oct 16 2013

Keywords

Examples

			1/(1*2*3*4) + 1/(2*3*4*5) + 1/(3*4*5*6) = 19/360, so a(3) = 19.
The rationals r(n) = a(n)/A230340(n) begin: 0, 1/24, 1/20, 19/360, 17/315, 55/1008, 83/1512, 119/2160, 82/1485, 73/1320, 95/1716, 121/2184, 227/4095, 559/10080, 679/12240, 815/14688, ... - _Wolfdieter Lang_, Mar 08 2018

References

  • L. B. W. Jolley, Summation of Series, Second revised ed., Dover, 1961, p.38, (202) and (201).

Links

Crossrefs

Cf. A001563, A052762, A094258, A125650, A230328, A230340 (denominators).

Programs

  • Mathematica
    a[n_] := Numerator[1/18 - 1/(3*(n+1)*(n+2)*(n+3))]; Table[a[n], {n, 0, 100}]
  • PARI
    a(n) = numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) \\ Colin Barker, Jul 30 2019

Formula

Numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) (from the generic formula Sum_{k=1..n} 1/Pochhammer(k, m) = 1/((m-1)*(m-1)!) - 1/((m-1)*Pochhammer(n+1, m-1)) with m = 4).
G.f. for the rationals r(n) = (1/18)*n*(11+n^2+6*n)/((1+n)*(n+2)*(n+3)) = a(n)/A230340(n): (1/18)*(1 - hypergeometric([1, 3], [4], -x/(1-x)))/(1-x) = (6*x - 15*x^2 + 11*x^3 + 6*(1 - 3*x + 3*x^2 - x^3)*log(1-x))/(36*x^3*(1-x)). - Wolfdieter Lang, Mar 08 2018
a(n) = numerator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))). - Colin Barker, Jul 30 2019

A230340 Denominator of Sum_{k=1..n} 1/(k(k+1)(k+2)(k+3)) = Sum_{k=1..n} 1/Pochhammer(k,4).

Original entry on oeis.org

1, 24, 20, 360, 315, 1008, 1512, 2160, 1485, 1320, 1716, 2184, 4095, 10080, 12240, 14688, 8721, 20520, 7980, 3080, 5313, 36432, 41400, 46800, 26325, 58968, 65772, 8120, 13485, 9920, 98208, 107712, 58905, 128520, 139860, 151848, 27417, 59280
Offset: 0

Views

Author

Jean-François Alcover, Oct 16 2013

Keywords

Examples

			1/(1*2*3*4) + 1/(2*3*4*5) + 1/(3*4*5*6) = 19/360, so a(3) = 360.

Links

Crossrefs

Cf. A001563, A052762, A094258, A125650, A230328, A230339 (numerators).

Programs

  • Magma
    [Denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))):n in [0..100]]; // Marius A. Burtea, Jul 30 2019
  • Mathematica
    a[n_] := Denominator[1/18 - 1/(3*(n+1)*(n+2)*(n+3))]; Table[a[n], {n, 0, 100}]
  • PARI
    a(n) = denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))) \\ Colin Barker, Jul 30 2019

Formula

Denominator(1/18 - 1/(3*(n+1)*(n+2)*(n+3))).

A300298 Numerators of r(n) := Sum_{k=0..n-1} 1/Product_{j=0..4} (k + j + 1), for n >= 0, with r(0) = 0.

Original entry on oeis.org

0, 1, 7, 17, 23, 125, 209, 329, 247, 119, 125, 341, 1819, 793, 3059, 3875, 1211, 187, 1219, 4427, 10625, 12649, 4983, 17549, 10237, 11875, 6851, 1311, 35959, 40919, 46375, 17453, 7363, 16511, 36907, 41125, 30463, 101269, 111929, 123409
Offset: 0

Views

Author

Wolfdieter Lang, Apr 05 2018

Keywords

Comments

The denominators are given in A300299.
The sum given in the name is computed using a telescopic sum. See the general recipe given in the Jolley reference, (201), p. 38.

Examples

			The sum begins: 0 + 1/(1*2*3*4*5) + 1/(2*3*4*5*6) + ... =  0 + 1/120 + 1/720 + 1/2520 + 1/6720 + 1/15120 + 1/30240 + ...
The rationals r(n) (partial sums) begin: 0/1, 1/120, 7/720, 17/1680, 23/2240, 125/12096, 209/20160, 329/31680, 247/23760, 119/11440, 125/12012, 341/32760, ...

References

  • L. B. W. Jolley, Summation of Series, Dover Publications, 2nd rev. ed., 1961, p. 38, (201).

Crossrefs

Cf. A230339/A230328, A300299.

Programs

  • GAP
    List(List([0..40],n->Sum([0..n-1],k->1/(Product([0..4],j->k+j+1)))),NumeratorRat); # Muniru A Asiru, Apr 05 2018
  • Magma
    [Numerator(n*(50+35*n+10*n^2+n^3)/(96*(1+n)*(2+n)*(n+3)*(4+n))): n in [0..50]]; // Vincenzo Librandi, Apr 06 2018
  • Mathematica
    Table[Numerator[n (50 + 35 n + 10 n^2 + n^3) / (96 (1 + n) (2 + n) (n + 3) (4 + n))], {n, 0, 50}] (* Vincenzo Librandi, Apr 06 2018 *)
  • PARI
    a(n) = numerator(sum(k=0, n-1, prod(j=0, 4, (k+j+1))^(-1))); \\ Altug Alkan, Apr 05 2018

Formula

a(n) = numerator(r(n)), with the result of the sum given in the name r(n) = n*(50 + 35*n + 10*n^2 + n^3)/(96*(1 + n)*(2 + n)*(n + 3)*(4 + n)), n >= 0.
This results from r(n) = 1/96 - 1/(4*(1+n)*(2+n)*(n+3)*(4+n)).
G.f. for rationals {r(n)}_{n >= 0}: (1/96)*(1 - hypergeometric([1, 4], [5], -x/(1-x)))/(1-x)
= (-x*(12 - 42*x + 52*x^2 - 25*x^3) + 12*(1 - x)^4*log(1/(1-x))) / (288*x^4*(1-x)).
Showing 1-4 of 4 results.

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