A214401 -id:A214401 - cp's OEIS frontend

A120266 Numerator of Sum_{k=0..n} n^k/k!.

2, 5, 13, 103, 1097, 1223, 47273, 556403, 10661993, 7281587, 62929017101, 7218065, 60718862681977, 595953719897, 13324966405463, 247016301114823, 28505097599389815853, 549689343118061, 320305944459287485595917

Offset: 1

Alexander Adamchuk, Jun 30 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 2, 5, 13, 103/3, 1097/12, 1223/5, 47273/72, 556403/315, 10661993/2240, ... = A120266/A214401. - _Petros Hadjicostas_, May 12 2020

Eric Weisstein, Exponential Sum Function.

Denominators are A214401. Cf. also A063170, A090878, A119029, A120267, A214402.

Numerator[Table[Sum[n^k/k!, {k,0,n}], {n,1,30}]]

a(n) = numerator(Sum_{k=0..n} n^k/k!).

a(n) = A063170(n)/A214402(n) = (n!/A214402(n))*Sum_{k=0..n} n^k/k! for n > 0. - Jonathan Sondow, Jul 16 2012

A119029 Numerator of Sum_{k=1..n} n^(k-1)/k!.

Original entry on oeis.org

1, 2, 4, 25, 217, 203, 6743, 69511, 1184417, 728102, 5720654791, 601499, 4670663321629, 42568060798, 888330615353, 15438515749903, 1676770323947695709, 30538296012677, 16858207434636875406943

Offset: 1

Alexander Adamchuk, Jul 22 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 1, 2, 4, 25/3, 217/12, 203/5, 6743/72, 69511/315, 1184417/2240, 728102/567, ... = A119029/A214401. - _Petros Hadjicostas_, May 12 2020

Cf. A063170, A090878, A093101, A120266, A120267, A214401 (denominators), A214402.

Numerator[Table[Sum[n^(k-1)/k!,{k,1,n}],{n,1,30}]]

a(n) = numerator(Sum_{k=1..n} n^(k-1)/k!).

a(n) = A120267(n)/n.

A120267 Numerator of Sum_{k=1..n} n^k/k!.

Original entry on oeis.org

1, 4, 12, 100, 1085, 1218, 47201, 556088, 10659753, 7281020, 62927202701, 7217988, 60718623181177, 595952851172, 13324959230295, 247016251998448, 28505095507110827053, 549689328228186, 320305941258100632731917

Offset: 1

Alexander Adamchuk, Jun 30 2006

n divides a(n) and a(n)/n = A119029(n). - Alexander Adamchuk, Oct 08 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 1, 4, 12, 100/3, 1085/12, 1218/5, 47201/72, 556088/315, 10659753/2240, 7281020/567, ... = A120267/A214401. - _Petros Hadjicostas_, May 12 2020

Cf. A063170, A090878, A093101, A119029, A120266, A214401 (denominators), A214402.

Numerator[Table[Sum[n^k/k!, {k,1,n}], {n,1,30}]]

a(n) = numerator(Sum_{k=1..n} n^k/k!).

a(n) = n*A119029(n). - Alexander Adamchuk, Oct 08 2006

Various sections edited by Petros Hadjicostas, May 12 2020~

A214402 Cancellation factor in reducing Sum_{k=0...n} n^k/k! to lowest terms.

Original entry on oeis.org

1, 2, 6, 8, 10, 144, 70, 128, 162, 6400, 22, 6220800, 26, 100352, 182250, 425984, 170, 429981696, 38, 163840000, 13502538, 317194240, 46, 247669456896, 31250, 1417674752, 15943230, 80564191232, 9802, 25076532510720000000, 62, 10737418240, 38196790434, 1241245548544

Offset: 1

Jonathan Sondow, Jul 15 2012

nonn

Michel Marcus, Table of n, a(n) for n = 1..300
Eric Weisstein, Exponential Sum Function.

Cf. A063170, A090878, A093101, A120266, A214401.

Table[n!/Denominator[Sum[n^k/k!, {k, 0, n}]], {n, 1, 30}]

a(n) = n!/denominator(sum(k=0, n, n^k/k!)); \\ Michel Marcus, Apr 20 2021

a(n) = n!/A214401(n).

More terms from Michel Marcus, Apr 20 2021

cp's OEIS Frontend

A120266 Numerator of Sum_{k=0..n} n^k/k!.

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A119029 Numerator of Sum_{k=1..n} n^(k-1)/k!.

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A120267 Numerator of Sum_{k=1..n} n^k/k!.

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A214402 Cancellation factor in reducing Sum_{k=0...n} n^k/k! to lowest terms.

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