A118391 -id:A118391 - cp's OEIS frontend

A118392 Denominator of sum of reciprocals of first n tetrahedral numbers A000292.

1, 4, 20, 5, 7, 56, 24, 15, 55, 44, 52, 91, 35, 80, 272, 51, 57, 380, 140, 77, 253, 184, 200, 325, 117, 252, 812, 145, 155, 992, 352, 187, 595, 420, 444, 703, 247, 520, 1640, 287, 301, 1892, 660, 345, 1081, 752, 784, 1225, 425, 884, 2756, 477, 495, 3080

Offset: 1

Jonathan Vos Post, Apr 27 2006

Numerators are A118391.

			a(1) = 1 = denominator of 1/1.
a(2) = 4 = denominator of 5/4 = 1/1 + 1/4.
a(3) = 20 = denominator of 27/20 = 1/1 + 1/4 + 1/10.
a(4) = 5 = denominator of 7/5 = 1/1 + 1/4 + 1/10 + 1/20.
a(5) = 7 = denominator of 10/7 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35.
a(20) = 77 = denominator of 115/77 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35 + 1/56 + 1/84 + 1/120 + 1/165 + 1/220 + 1/286 + 1/364 + 1/455 + 1/560 + 1/680 + 1/816 + 1/969 + 1/1140 + 1/1330 + 1/1540.
Fractions are: 1/1, 5/4, 27/20, 7/5, 10/7, 81/56, 35/24, 22/15, 81/55, 65/44, 77/52, 135/91, 52/35, 119/80, 405/272, 76/51, 85/57, 567/380, 209/140, 115/77, 378/253, 275/184, 299/200, 486/325, 175/117, 377/252, 1215/812, 217/145, 232/155, 1485/992.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000292, A022998, A026741, A118391.

[Denominator(3*n*(n+3)/(2*(n+1)*(n+2))): n in [1..60]]; // G. C. Greubel, Feb 18 2021

A118392:= n -> denom(3*n*(n+3)/(2*(n+1)*(n+2)));
seq(A118392(n), n = 1..60); # G. C. Greubel, Feb 18 2021

Accumulate[1/Binomial[Range[70]+2,3]]//Denominator (* Harvey P. Dale, Jun 07 2018 *)

s=0;for(i=3,50,s+=1/binomial(i,3);print(denominator(s))) /* Phil Carmody, Mar 27 2012 */

[denominator(3*n*(n+3)/(2*(n+1)*(n+2))) for n in (1..60)] # G. C. Greubel, Feb 18 2021

A118391(n)/A118392(n) = Sum_{i=1..n} 1/A000292(n).
A118391(n)/A118392(n) = Sum_{i=1..n} 1/C(n+2,3).
A118391(n)/A118392(n) = Sum_{i=1..n} 6/(n*(n+1)*(n+2)).
a(n) = denominator( 3*n*(n+3)/(2*(n+1)*(n+2)) ). - G. C. Greubel, Feb 18 2021

More terms from Harvey P. Dale, Jun 07 2018

A118411 Numerator of sum of reciprocals of first n pentatope numbers A000332.

Original entry on oeis.org

1, 6, 19, 136, 83, 119, 656, 73, 190, 121, 1816, 559, 679, 815, 3872, 1139, 886, 513, 2360, 2023, 2299, 2599, 11696, 3275, 7306, 1353, 5992, 1653, 5455, 5983, 26176, 7139, 15538, 8435, 12184, 3293, 3553, 11479, 49360

Offset: 1

Jonathan Vos Post, Apr 27 2006

Denominators are A118412. Fractions are: 1/1, 6/5, 19/15, 136/105, 83/63, 119/90, 656/495, 73/55, 190/143, 121/91, 1816/1365, 559/420, 679/510, 815/612, 3872/2907, 1139/855, 886/665, 513/385, 2360/1771, 2023/1518, 2299/1725, 2599/1950, 11696/8775, 3275/2457, 7306/5481, 1353/1015, 5992/4495, 1653/1240, 5455/4092, 5983/4488, 26176/19635, 7139/5355, 15538/11655, 8435/6327, 12184/9139, 3293/2470, 3553/2665, 11479/8610, 49360/37023. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392.

			a(1) = 1 = numerator of 1/1.
a(2) = 6 = numerator of 6/5 = 1/1 + 1/5.
a(3) = 19 = numerator of 19/15 = 1/1 + 1/5 + 1/15.
a(4) = 136 = numerator of 136/105 = 1/1 + 1/5 + 1/15 + 1/35.
a(5) = 55 = numerator of 55/42 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70.
a(10) = 190 = numerator of 190/143 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715.
a(20) = 2360 = numerator of 2360/1771 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715 + 1/1001 + 1/1365 + 1/1820 + 1/2380 + 1/3060 + 1/3876 + 1/4845 + 1/5985 + 1/7315 + 1/8855.

Cf. A000332, A022998, A026741, A118391, A118391, A118412.

s=0;for(i=4,50,s+=1/binomial(i,4);print(numerator(s))) /* Phil Carmody, Mar 27 2012 */

A118411(n)/A118412(n) = SUM[i=1..n] (1/A000332(n)). A118411(n)/A118412(n) = SUM[i=1..n] (1/C(n+2,4)). A118411(n)/A118412(n) = SUM[i=1..n] (1/(n*(n+1)*(n+2)*(n+3)/24)).

A118412 Denominator of sum of reciprocals of first n pentatope numbers A000332.

Original entry on oeis.org

1, 5, 15, 105, 42, 63, 90, 495, 55, 143, 91, 1365, 420, 510, 612, 2907, 855, 665, 385, 1771, 1518, 1725, 1950, 8775, 2457, 5481, 1015, 4495, 1240, 4092, 4488, 19635, 5355, 11655, 6327, 9139, 2470, 2665, 8610, 37023

Offset: 1

Jonathan Vos Post, Apr 27 2006

Numerators are A118411. Fractions are: 1/1, 6/5, 19/15, 136/105, 83/63, 119/90, 656/495, 73/55, 190/143, 121/91, 1816/1365, 559/420, 679/510, 815/612, 3872/2907, 1139/855, 886/665, 513/385, 2360/1771, 2023/1518, 2299/1725, 2599/1950, 11696/8775, 3275/2457, 7306/5481, 1353/1015, 5992/4495, 1653/1240, 5455/4092, 5983/4488, 26176/19635, 7139/5355, 15538/11655, 8435/6327, 12184/9139, 3293/2470, 3553/2665, 11479/8610, 49360/37023. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392.

			a(1) = 1 = denominator of 1/1.
a(2) = 5 = denominator of 6/5 = 1/1 + 1/5.
a(3) = 15 = denominator of 19/15 = 1/1 + 1/5 + 1/15.
a(4) = 105 = denominator of 136/105 = 1/1 + 1/5 + 1/15 + 1/35.
a(5) = 42 = denominator of 55/42 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70.
a(10) = 143 = denominator of 190/143 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715.
a(20) = 1771 = denominator of 2360/1771 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715 + 1/1001 + 1/1365 + 1/1820 + 1/2380 + 1/3060 + 1/3876 + 1/4845 + 1/5985 + 1/7315 + 1/8855.

Cf. A000332, A022998, A026741, A118391, A118391, A118411.

s=0;for(i=4,50,s+=1/binomial(i,4);print(denominator(s))) /* Phil Carmody, Mar 27 2012 */

A118411(n)/A118412(n) = SUM[i=1..n] (1/A000332(n)). A118411(n)/A118412(n) = SUM[i=1..n] (1/C(n+2,4)). A118411(n)/A118412(n) = SUM[i=1..n] (1/(n*(n+1)*(n+2)*(n+3)/24)).

A118431 Numerator of sum of reciprocals of first n 5-simplex numbers A000389.

Original entry on oeis.org

1, 7, 17, 69, 625, 209, 329, 247, 357, 1250, 341, 1819, 2379, 3059, 19375, 1211, 1496, 3657, 4427, 53125, 12649, 14949, 17549, 10237, 59375, 6851, 7866, 35959, 40919, 231875, 52359, 14726, 16511, 36907, 205625, 91389, 101269

Offset: 1

Jonathan Vos Post, Apr 28 2006

Denominators are A118432. Fractions are: 1/1, 7/6, 17/14, 69/56, 625/504, 209/168, 329/264, 247/198, 357/286, 1250/1001, 341/273, 1819/1456, 2379/1904, 3059/2448, 19375/15504, 1211/969, 1496/1197, 3657/2926, 4427/3542, 53125/42504, 12649/10120, 14949/11960, 17549/14040, 10237/8190, 59375/47502, 6851/5481, 7866/6293, 35959/28768, 40919/32736, 231875/185504, 52359/41888, 14726/11781, 16511/13209, 36907/29526, 205625/164502, 91389/73112, 101269/81016. The numerator of sum of reciprocals of first n triangular numbers is A026741. The numerator of sum of reciprocals of first n tetrahedral numbers is A118391. The numerator of sum of reciprocals of first n pentatope numbers is A118411.

			a(1) = 1 = numerator of 1/1.
a(2) = 7 = numerator of 7/6 = 1/1 + 1/6.
a(3) = 17 = numerator of 17/14 = 1/1 + 1/6 + 1/21.
a(4) = 69 = numerator of 69/56 = 1/1 + 1/6 + 1/21 + 1/56.
a(5) = 55 = numerator of 55/42 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126.
a(10) = 1250 = numerator of 1250/1001 = 1/1+ 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002.
a(20) = 53125 = numerator of 53125/42504 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002 + 1/3003 + 1/4368 + 1/6188 + 1/8568 + 1/11628 + 1/15504 + 1/20349 + 1/26334 + 1/33649 + 1/42504.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000332, A000389, A022998, A026741, A118391, A118391, A118411, A118412, A118432.

Numerator[Accumulate[1/Binomial[Range[5, 50], 5]]] (* G. C. Greubel, Nov 21 2017 *)

A118411(n)/A118412(n) = Sum_{i=1..n} (1/A000389(n)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/C(n,5)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/(n*(n+1)*(n+2)*(n+3)*(n+4)/120)).

A118432 Denominator of sum of reciprocals of first n 5-simplex numbers A000389.

Original entry on oeis.org

1, 6, 14, 56, 504, 168, 264, 198, 286, 1001, 273, 1456, 1904, 2448, 15504, 969, 1197, 2926, 3542, 42504, 10120, 11960, 14040, 8190, 47502, 5481, 6293, 28768, 32736, 185504, 41888, 11781, 13209, 29526, 164502, 73112, 81016

Offset: 1

Jonathan Vos Post, Apr 28 2006

Numerators are A118431. Fractions are: 1/1, 7/6, 17/14, 69/56, 625/504, 209/168, 329/264, 247/198, 357/286, 1250/1001, 341/273, 1819/1456, 2379/1904, 3059/2448, 19375/15504, 1211/969, 1496/1197, 3657/2926, 4427/3542, 53125/42504, 12649/10120, 14949/11960, 17549/14040, 10237/8190, 59375/47502, 6851/5481, 7866/6293, 35959/28768, 40919/32736, 231875/185504, 52359/41888, 14726/11781, 16511/13209, 36907/29526, 205625/164502, 91389/73112, 101269/81016. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392. The denominator of sum of reciprocals of first n pentatope numbers is A118412.

			a(1) = 1 = denominator of 1/1.
a(2) = 6 = denominator of 7/6 = 1/1 + 1/6.
a(3) = 14 = denominator of 17/14 = 1/1 + 1/6 + 1/21.
a(4) = 56 = denominator of 69/56 = 1/1 + 1/6 + 1/21 + 1/56.
a(5) = 42 = denominator of 55/42 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126.
a(10) = 1001 = denominator of 1250/1001 = 1/1+ 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002.
a(20) = 42504 = denominator of 53125/42504 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002 + 1/3003 + 1/4368 + 1/6188 + 1/8568 + 1/11628 + 1/15504 + 1/20349 + 1/26334 + 1/33649 + 1/42504.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000332, A000389, A022998, A026741, A118391, A118391, A118411, A118412, A118431.

Denominator[Accumulate[1/Binomial[Range[5,50],5]]] (* Harvey P. Dale, Jul 17 2016 *)

A118411(n)/A118412(n) = Sum_{i=1..n} (1/A000389(n)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/C(n,5)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/(n*(n+1)*(n+2)*(n+3)*(n+4)/120)).

A118583 Numerator of sum of first p reciprocals of p-simplex numbers divided by p^4, where p = prime(n) for n > 2.

Original entry on oeis.org

1, 5, 53, 789, 237493, 2576561, 338350897, 616410400171, 2603853251291, 5745400286707685, 3081677433937346539, 41741941495866750557, 7829195555633964779233, 21066131970056662377432067

Offset: 3

Alexander Adamchuk, May 09 2007

nonn

			Prime(3) = 5.
a(3) = 1 because A118431(5)/5^4 = 1, where A118431(5) = Numerator[ 1/C(4+1,5) + 1/C(4+2,5) + 1/C(4+3,5) + 1/C(4+4,5) +1/C(4+5,5) ] = Numerator[ 1/1 + 1/6 + 1/21 + 1/56 + 1/126 ] = 625.

G. C. Greubel, Table of n, a(n) for n = 3..250
Eric Weisstein's World of Mathematics, Composition.
Eric Weisstein's World of Mathematics, Tetrahedral Number.
Eric Weisstein's World of Mathematics, Triangular Number.

Cf. A022998 = Numerator of sum of reciprocals of first n triangular numbers
Cf. A118391 = Numerator of sum of reciprocals of first n tetrahedral numbers A000292.
Cf. A118431 = Numerator of sum of reciprocals of first n 5-simplex numbers A000389.

Table[Numerator[Sum[1 /Binomial[ n+Prime[k]-1, Prime[k]], {n,1,Prime[k]} ]]/ Prime[k]^4, {k,3,25}]

for(n=3,10, print1(numerator(sum(k=1,prime(n), 1/(binomial(k+ prime(n)-1, prime(n)))))/prime(n)^4, ", ")) \\ G. C. Greubel, Nov 25 2017

a(n) = numerator( Sum_{k=1..prime(n)} ( 1/binomial( k + prime(n) - 1, prime(n) ) ))/prime(n)^4 for n > 2.

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A118392 Denominator of sum of reciprocals of first n tetrahedral numbers A000292.

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A118411 Numerator of sum of reciprocals of first n pentatope numbers A000332.

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A118412 Denominator of sum of reciprocals of first n pentatope numbers A000332.

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A118431 Numerator of sum of reciprocals of first n 5-simplex numbers A000389.

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A118432 Denominator of sum of reciprocals of first n 5-simplex numbers A000389.

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A118583 Numerator of sum of first p reciprocals of p-simplex numbers divided by p^4, where p = prime(n) for n > 2.

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