A254028 -id:A254028 - cp's OEIS frontend

A254030 a(n) = 14^n + 23^n + 32^n + 41^n.

10, 20, 50, 146, 470, 1610, 5750, 21146, 79430, 303050, 1169750, 4554746, 17852390, 70322090, 278050550, 1102537946, 4381257350, 17438542730, 69495104150, 277204002746, 1106488342310, 4418973508970, 17654960746550

Offset: 0

Luciano Ancora, Jan 26 2015

This is the sequence of fourth terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A052548, A254028, A254031, A254144, A254145, A254146.

seq(add(i*(5 - i)^n, i = 1..4), n = 0..20); # Peter Bala, Jan 31 2017

Table[3 2^n + 2^(2 n) + 2 3^n + 4, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)
LinearRecurrence[{10,-35,50,-24},{10,20,50,146},30] (* Harvey P. Dale, Jun 06 2020 *)

Vec(-2*(77*x^3-100*x^2+40*x-5)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1))  + O(x^100)) \\ Colin Barker, Jan 26 2015

G.f.: -2*(77*x^3-100*x^2+40*x-5) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 4.
a(n) = 1/3!*Sum_{k = 0..n} (-1)^(k+n)*(k + 5)!*Stirling2(n,k)/
((k + 1)*(k + 2)). (End)
E.g.f.: exp(x)*(4 + 3*exp(x) + 2*exp(2*x) + exp(3*x)). - Stefano Spezia, May 19 2025

A254031 a(n) = 15^n + 24^n + 33^n + 42^n + 5*1^n.

Original entry on oeis.org

15, 35, 105, 371, 1449, 6035, 26265, 117971, 542409, 2538515, 12044025, 57756371, 279305769, 1359736595, 6654800985, 32708239571, 161307227529, 797687136275, 3953299529145, 19626731023571, 97576919443689, 485664640673555

Offset: 0

Luciano Ancora, Jan 26 2015

This is the sequence of fifth terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Cf. A052548, A254028, A254030, A254144, A254145, A254146.

seq(add(i*(6 - i)^n, i = 1..5), n = 0..20); # Peter Bala, Jan 31 2017

Table[2^(n + 2) + 2^(2 n + 1) + 3^(n + 1) + 5^n + 5, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)
LinearRecurrence[{15,-85,225,-274,120},{15,35,105,371,1449},30] (* Harvey P. Dale, Jan 24 2022 *)

Vec(-(1044*x^4-1604*x^3+855*x^2-190*x+15)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)) + O(x^100)) \\ Colin Barker, Jan 26 2015

G.f.: -(1044*x^4 - 1604*x^3 + 855*x^2 - 190*x + 15) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 5.
a(n) = (1/4!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 6)!*Stirling2(n,k)/
((k + 1)*(k + 2)). (End)

A254144 a(n) = 16^n + 25^n + 34^n + 43^n + 52^n + 61^n.

Original entry on oeis.org

21, 56, 196, 812, 3724, 18236, 93436, 494732, 2685004, 14851676, 83384476, 473755052, 2717541484, 15709845116, 91395715516, 534498925772, 3139343105164, 18504595174556, 109397060622556, 648335998054892, 3850205790608044

Offset: 0

Luciano Ancora, Jan 26 2015

This is the sequence of sixth terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A052548, A254028, A254030, A254031, A254145, A254146.

seq(add(i*(7 - i)^n, i = 1..6), n = 0..20); # Peter Bala, Jan 31 2017

Table[5 2^n + 3 4^n + 4 3^n + 2 5^n + 6^n + 6, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)

Vec(-(8028*x^5-13916*x^4+8939*x^3-2695*x^2+385*x-21)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Jan 26 2015

G.f.: -(8028*x^5 - 13916*x^4 + 8939*x^3 - 2695*x^2 + 385*x - 21) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 6.
a(n) = (1/5!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 7)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)

A254145 a(n) = 17^n + 26^n + 35^n + 44^n + 53^n + 62^n + 7*1^n.

Original entry on oeis.org

28, 84, 336, 1596, 8400, 47244, 278256, 1695036, 10592400, 67518444, 437200176, 2867080476, 18997064400, 126948964044, 854359702896, 5783851121916, 39350309552400, 268842017200044, 1843254419626416, 12675940450459356

Offset: 0

Luciano Ancora, Jan 26 2015

This is the sequence of seventh terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formula
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A052548, A254028, A254030, A254031, A254144, A254146.

seq(add(i*(8 - i)^n, i = 1..7), n = 0..20); # Peter Bala, Jan 31 2017

Table[6 2^n + 4 4^n + 5 3^n + 2 6^n + 3 5^n + 7^n + 7, {n, 0, 25}] (*  *)
LinearRecurrence[{28,-322,1960,-6769,13132,-13068,5040},{28,84,336,1596,8400,47244,278256},30] (* or *) Table[Total[ Range[ 7]Range[ 7,1,-1]^n],{n,0,20}] (* Harvey P. Dale, Jun 21 2016 *)

Vec(-4*(17316*x^6 -32926*x^5 +24199*x^4 -8911*x^3 +1750*x^2 -175*x +7) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1))  + O(x^100)) \\ Colin Barker, Jan 26 2015

G.f.: -4*(17316*x^6 - 32926*x^5 + 24199*x^4 - 8911*x^3 + 1750*x^2 - 175*x + 7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 7.
a(n) = (1/6!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 8)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)

A254146 a(n) = 18^n + 27^n + 36^n + 45^n + 54^n + 63^n + 72^n + 81^n.

Original entry on oeis.org

36, 120, 540, 2892, 17172, 109020, 725220, 4992492, 35277012, 254402940, 1864757700, 13850340492, 103996064052, 787943896860, 6015370201380, 46217575406892, 357036252710292, 2770979252910780, 21591510288112260, 168818732978719692, 1323861500735007732

Offset: 0

Luciano Ancora, Jan 27 2015

This is the sequence of eighth terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Cf. A052548, A254028, A254030, A254031, A254144, A254145.

[7*2^n+5*4^n+8^n+6*3^n+3*6^n+4*5^n+2*7^n+8: n in [0..30]]; // Vincenzo Librandi, Jan 28 2015

seq(add(i*(9 - i)^n, i = 1..8), n = 0..20); # Peter Bala, Jan 31 2017

Table[7 2^n + 5 4^n + 8^n + 6 3^n + 3 6^n + 4 5^n + 2 7^n + 8, {n, 0, 30}] (* Vincenzo Librandi, Jan 28 2015 *)
LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{36,120,540,2892,17172,109020,725220,4992492},30] (* Harvey P. Dale, Mar 02 2022 *)

vector(30, n, n--; 7*2^n + 5*4^n + 8^n + 6*3^n + 3*6^n + 4*5^n + 2*7^n + 8) \\ Colin Barker, Jan 28 2015

G.f.: -12*(55308*x^7 - 113262*x^6 + 92327*x^5 - 39312*x^4 + 9527*x^3 - 1323*x^2 + 98*x -3) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Jan 28 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 8.
a(n) = 1/7!*Sum_{k = 0..n} (-1)^(k+n)*(k + 9)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)
a(n) = 36*a(n-1)-546*a(n-2)+4536*a(n-3)-22449*a(n-4)+67284*a(n-5)-118124*a(n-6)+109584*a(n-7)-40320*a(n-8). - Wesley Ivan Hurt, May 24 2021

A254147 a(n) = 19^n + 28^n + 37^n + 46^n + 55^n + 64^n + 73^n + 82^n + 9*1^n.

Original entry on oeis.org

45, 165, 825, 4917, 32505, 229845, 1703625, 13072917, 103008345, 828707925, 6779099625, 56214660117, 471424600185, 3990804658005, 34053173154825, 292542431786517, 2527742384720025, 21950298188288085, 191434401453597225, 1675813243179972117

Offset: 0

Luciano Ancora, Jan 28 2015

This is the sequence of ninth terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

Cf. A052548, A254028, A254030, A254031, A254144, A254145, A254146.

seq(add(i*(10-i)^n, i = 1..9), n = 0..20); # Peter Bala, Jan 31 2017

vector(30, n, n--; 8*2^n + 6*4^n + 2*8^n + 7*3^n + 4*6^n + 9^n + 5*5^n + 3*7^n + 9) \\ Colin Barker, Jan 28 2015

G.f.: -3*(2333280*x^8 - 5080464*x^7 + 4500500*x^6 - 2143640*x^5 + 605675*x^4 - 104636*x^3 + 10850*x^2 - 620*x + 15) / ((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(7*x - 1)*(8*x - 1)*(9*x - 1)). - Colin Barker, Jan 28 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 9.
a(n) = (1/8!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 10)!*Stirling2(n,k) /((k + 1)*(k + 2)). (End)

A254148 a(n) = 92^n + 74^n + 38^n + 83^n + 29^n + 65^n + 56^n + 47^n + 10^n + 10.

Original entry on oeis.org

55, 220, 1210, 7942, 57838, 450670, 3682030, 31153342, 270739678, 2403012910, 21693441550, 198578979742, 1838853136318, 17193665419150, 162090976108270, 1538867288166142, 14698448516729758, 141129617123665390, 1361277292619082190

Offset: 0

Luciano Ancora, Jan 28 2015

This is the sequence of tenth terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..999
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Cf. A052548, A254028, A254030, A254031, A254144, A254145, A254146, A254147.

vector(30, n, n--; 9*2^n + 7*4^n + 3*8^n + 8*3^n + 2*9^n + 6*5^n + 5*6^n + 4*7^n + 10^n + 10) \\ Colin Barker, Jan 28 2015

G.f.: -(80627040*x^9 -184920912*x^8 +175484892*x^7 -91478420*x^6 +29111445*x^5 -5902743*x^4 +766458*x^3 -61710*x^2 +2805*x -55) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)). - Colin Barker, Jan 28 2015

cp's OEIS Frontend

A254030 a(n) = 1*4^n + 2*3^n + 3*2^n + 4*1^n.

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A254031 a(n) = 1*5^n + 2*4^n + 3*3^n + 4*2^n + 5*1^n.

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A254144 a(n) = 1*6^n + 2*5^n + 3*4^n + 4*3^n + 5*2^n + 6*1^n.

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A254145 a(n) = 1*7^n + 2*6^n + 3*5^n + 4*4^n + 5*3^n + 6*2^n + 7*1^n.

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A254146 a(n) = 1*8^n + 2*7^n + 3*6^n + 4*5^n + 5*4^n + 6*3^n + 7*2^n + 8*1^n.

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A254147 a(n) = 1*9^n + 2*8^n + 3*7^n + 4*6^n + 5*5^n + 6*4^n + 7*3^n + 8*2^n + 9*1^n.

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A254148 a(n) = 9*2^n + 7*4^n + 3*8^n + 8*3^n + 2*9^n + 6*5^n + 5*6^n + 4*7^n + 10^n + 10.

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A254030 a(n) = 14^n + 23^n + 32^n + 41^n.

A254031 a(n) = 15^n + 24^n + 33^n + 42^n + 5*1^n.

A254144 a(n) = 16^n + 25^n + 34^n + 43^n + 52^n + 61^n.

A254145 a(n) = 17^n + 26^n + 35^n + 44^n + 53^n + 62^n + 7*1^n.

A254146 a(n) = 18^n + 27^n + 36^n + 45^n + 54^n + 63^n + 72^n + 81^n.

A254147 a(n) = 19^n + 28^n + 37^n + 46^n + 55^n + 64^n + 73^n + 82^n + 9*1^n.

A254148 a(n) = 92^n + 74^n + 38^n + 83^n + 29^n + 65^n + 56^n + 47^n + 10^n + 10.