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.

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A254470 Sixth partial sums of fourth powers (A000583).

Original entry on oeis.org

1, 22, 198, 1134, 4884, 17226, 52338, 141570, 348777, 795652, 1701700, 3444948, 6651216, 12321804, 22011804, 38073948, 63985977, 104782986, 167620090, 262495090, 403165620, 608300550, 902911230, 1320114510, 1903286385, 2708672616, 3808530792, 5294887048
Offset: 1

Views

Author

Luciano Ancora, Feb 15 2015

Keywords

Examples

			First differences:   1, 15,  65, 175,  369,   671, ... (A005917)
-------------------------------------------------------------------------
The fourth powers:   1, 16,  81, 256,  625,  1296, ... (A000583)
-------------------------------------------------------------------------
First partial sums:  1, 17,  98, 354,  979,  2275, ... (A000538)
Second partial sums: 1, 18, 116, 470, 1449,  3724, ... (A101089)
Third partial sums:  1, 19, 135, 605, 2054,  5778, ... (A101090)
Fourth partial sums: 1, 20, 155, 760, 2814,  8592, ... (A101091)
Fifth partial sums:  1, 21, 176, 936, 3750, 12342, ... (A254681)
Sixth partial sums:  1, 22, 198,1134, 4884, 17226, ... (this sequence)

Links

Crossrefs

Cf. A000538, A000583, A005917, A101089, A101090, A101091, A254469, A254471, A254472, A254681.

Programs

  • Magma
    [n*(1+n)*(2+n)*(3+n)^2*(4+n)*(5+n)*(6+n)*(1+12*n+ 2*n^2)/302400: n in [1..30]]; // Vincenzo Librandi, Feb 15 2015
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n)^2 (4 + n) (5 + n) (6 + n) (1 + 12 n + 2 n^2)/302400,{n, 25}] (* or *) CoefficientList[Series[(- 1 - 11 x - 11 x^2 - x^3)/(- 1 + x)^11, {x, 0, 24}], x]
Nest[Accumulate,Range[30]^4,6] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,22,198,1134,4884,17226,52338,141570,348777,795652,1701700},30] (* Harvey P. Dale, Apr 23 2016 *)
  • PARI
    vector(50,n,n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(1 + 12*n + 2*n^2)/302400) \\ Derek Orr, Feb 19 2015

Formula

G.f.: (-x - 11*x^2 - 11*x^3 - x^4)/(- 1 + x)^11.
a(n) = n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(1 + 12*n + 2*n^2)/302400.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^4.
Sum_{n>=1} 1/a(n) = 3320303/2601 + 1400*Pi^2/17 + (8960/17)*sqrt(2/17)*Pi*cot(sqrt(17/2)*Pi). - Amiram Eldar, Jan 26 2022

A254471 Sixth partial sums of fifth powers (A000584).

Original entry on oeis.org

1, 38, 456, 3210, 16290, 65922, 225576, 677742, 1834755, 4559620, 10547888, 22958364, 47415108, 93547260, 177288240, 324223524, 574358277, 988774554, 1658764600, 2718164150, 4359769830, 6856910190, 10591453080, 16089775650, 24068499975, 35492110056
Offset: 1

Views

Author

Luciano Ancora, Feb 15 2015

Keywords

Examples

			First differences:   1, 31, 211,  781,  2101,  4651, ... (A022521)
-------------------------------------------------------------------------
The fifth powers:    1, 32, 243, 1024,  3125,  7776, ... (A000584)
-------------------------------------------------------------------------
First partial sums:  1, 33, 276, 1300,  4425, 12201, ... (A000539)
Second partial sums: 1, 34, 310, 1610,  6035, 18236, ... (A101092)
Third partial sums:  1, 35, 345, 1955,  7990, 26226, ... (A101099)
Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ... (A254644)
Fifth partial sums:  1, 37, 418, 2754, 13080, 49632, ... (A254682)
Sixth partial sums:  1, 38, 456, 3210, 16290, 65922, ... (this sequence)

Links

Crossrefs

Cf. A000539, A000584, A022521, A101092, A101099, A254469, A254470, A254472, A254644, A254682.

Programs

  • Magma
    [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(-29+54*n+ 81*n^2+24*n^3+2*n^4)/665280: n in [1..30]]; // Vincenzo Librandi, Feb 15 2015
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (- 29 + 54 n + 81 n^2 + 24 n^3 + 2 n^4)/665280, {n, 23}] (* or *) CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(- 1 + x)^12, {x, 0, 28}], x]
  • PARI
    vector(50,n,n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-29 + 54*n + 81*n^2 + 24*n^3 + 2*n^4)/665280) \\ Derek Orr, Feb 19 2015

Formula

G.f.: (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)/(- 1 + x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-29 + 54*n + 81*n^2 + 24*n^3 + 2*n^4)/665280.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^5.

A254472 Sixth partial sums of sixth powers (A001014).

Original entry on oeis.org

1, 70, 1134, 9870, 59220, 275562, 1063530, 3552978, 10577385, 28652260, 71725108, 167911380, 371057232, 779831820, 1568210220, 3032733564, 5663906745, 10251608346, 18037546450, 30931714450, 51814612980, 84952851750, 136562787270, 215565263550, 334584493425
Offset: 1

Views

Author

Luciano Ancora, Feb 15 2015

Keywords

Examples

			First differences:   1, 63,  665, 3367, 11529, ... (A022522)
--------------------------------------------------------------------------
The sixth powers:    1, 64,  729, 4096, 15625, ... (A001014)
--------------------------------------------------------------------------
First partial sums:  1, 65,  794, 4890, 20515, ... (A000540)
Second partial sums: 1, 66,  860, 5750, 26265, ... (A101093)
Third partial sums:  1, 67,  927, 6677, 32942, ... (A254640)
Fourth partial sums: 1, 68,  995, 7672, 40614, ... (A254645)
Fifth partial sums:  1, 69, 1064, 8736, 49350, ... (A254683)
Sixth partial sums:  1, 70, 1134, 9870, 59220, ... (this sequence)

Links

Crossrefs

Cf. A000540, A001014, A022522, A101093, A254469, A254470, A254471, A254640, A254645, A254683.

Programs

  • Magma
    [n*(1+n)*(2+n)*(3+n)^2*(4+n)*(5+n)*(6+n)*(-3+5*n+n^2)* (3+7*n+n^2)/665280: n in [1..30]]; // Vincenzo Librandi, Feb 15 2015
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n)^2 (4 + n) (5 + n) (6 + n) (- 3 + 5 n + n^2) (3 + 7 n + n^2)/665280, {n, 22}] (* or *) CoefficientList[Series[(- 1 - 57 x - 302 x^2 - 302 x^3 - 57 x^4 - x^5)/(- 1 + x)^13, {x, 0, 28}], x]
Nest[Accumulate,Range[30]^6,6] (* Harvey P. Dale, Oct 02 2015 *)
  • PARI
    vector(50,n,n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(-3 + 5*n + n^2)*(3 + 7*n + n^2)/665280) \\ Derek Orr, Feb 19 2015

Formula

G.f.: (-x - 57*x^2 - 302*x^3 - 302*x^4 - 57*x^5 - x^6)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(-3 + 5*n + n^2)*(3 + 7*n + n^2)/665280.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^6.
Sum_{n>=1} 1/a(n) = 25622179/76545 - 3080*Pi^2/81 + 149600*Pi*tan(sqrt(37)*Pi/2)/(243*sqrt(37)). - Amiram Eldar, Jan 27 2022

A254642 Third partial sums of eighth powers (A001016).

Original entry on oeis.org

1, 259, 7335, 86765, 629174, 3314178, 13906578, 49183590, 152191935, 422931613, 1075761505, 2540663307, 5633367740, 11829663860, 23692442292, 45516670332, 84278105421, 150996708135, 262656041515, 444856105561, 735419759634, 1189222877270
Offset: 1

Views

Author

Luciano Ancora, Feb 05 2015

Keywords

Examples

			First differences:   1, 255, 6305, 58975, 325089, ...(A022524)
--------------------------------------------------------------------
The eighth powers:   1, 256, 6561, 65536, 390625, ...(A001016)
--------------------------------------------------------------------
First partial sums:  1, 257, 6818, 72354, 462979, ...(A000542)
Second partial sums: 1, 258, 7076, 79430, 542409, ...(A253636)
Third partial sums:  1, 259, 7335, 86765, 629174, ...(this sequence)

Links

Crossrefs

Cf. A000542, A001016, A022524, A253636.

Programs

  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (3 + 2 n) (1 + 36 n - 69 n^2 + 45 n^4 + 18 n^5 + 2 n^6)/3960, {n, 22}]
Accumulate[Accumulate[Accumulate[Range[22]^8]]]
CoefficientList[Series[(1 + 247 x + 4293 x^2 + 15619 x^3 + 15619 x^4 + 4293 x^5 + 247 x^6 + x^7)/(- 1 + x)^12, {x, 0, 22}], x]
  • PARI
    a(n)=n*(1+n)*(2+n)*(3+n)*(3+2*n)*(1+36*n-69*n^2+45*n^4+18*n^5+2*n^6)/3960 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)/(- 1 + x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(3 + 2*n)*(1 + 36*n - 69*n^2 + 45*n^4 + 18*n^5 + 2*n^6)/3960.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + n^8.

A254646 Fourth partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 132, 2709, 26432, 168126, 804552, 3136014, 10459968, 30856839, 82407052, 202678203, 465069696, 1005729452, 2066218896, 4058958828, 7664805504, 13974953853, 24692818836, 42415687153, 71020845504, 116186669130, 186085891160, 292296070170, 450981236160, 684408934755
Offset: 1

Views

Author

Luciano Ancora, Feb 05 2015

Keywords

Examples

			First differences:   1, 127, 2059, 14197,  61741, ...  (A022523)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (this sequence)

Links

Crossrefs

Cf. A000541, A001015, A022523, A250212, A254641, A254645.

Programs

  • GAP
    List([1..30], n-> Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198); # G. C. Greubel, Aug 28 2019
  • Magma
    [Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198: n in [1..30]]; // G. C. Greubel, Aug 28 2019
  • Maple
    seq(binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198, n=1..30); # G. C. Greubel, Aug 28 2019
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (48 - 100 n - 89 n^2 + 160 n^3 + 140 n^4 + 36 n^5 + 3 n^6)/23760, {n, 20}] (* or *)
Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^7]]]] (* or *)
CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(- 1 + x)^12, {x, 0, 19}], x]
  • PARI
    a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(48-100*n-89*n^2+160*n^3+140*n^4 +36*n^5+3*n^6)/23760 \\ Charles R Greathouse IV, Oct 07 2015
  • Sage
    [binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198 for n in (1..30)] # G. C. Greubel, Aug 28 2019

Formula

G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(48 - 100*n - 89*n^2 + 160*n^3 + 140*n^4 + 36*n^5 + 3*n^6)/23760.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^7.

A254684 Fifth partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 133, 2842, 29274, 197400, 1001952, 4137966, 14597934, 45454773, 127861825, 330540028, 795609724, 1801339176, 3867558072, 7926516900, 15591322404, 29566276257, 54259095093, 96674782246, 167695627750, 283882296880
Offset: 1

Views

Author

Luciano Ancora, Feb 12 2015

Keywords

Examples

			First differences:   1, 127, 2060, 14324,  63801, ...  (A152726)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (A254646)
Fifth partial sums:  1, 133, 2842, 29274, 197400, ...  (this sequence)

Links

Crossrefs

Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254681, A254682, A254683.

Programs

  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 3 + 5 n + n^2) (- 2 + 5 n + n^2) (5 + 5 n + n^2)/95040, {n,21}] (* or *)
CoefficientList[Series[(- 1 - 120 x - 1191 x^2 - 2416 x^3 - 1191 x^4 - 120 x^5 - x^6)/(-1 + x)^13, {x,0,20}], x]
  • PARI
    a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-3+5*n+n^2)*(-2+5*n+n^2)*(5+5*n+n^2)/95040 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (- x - 120*x^2 - 1191*x^3 - 2416*x^4 - 1191*x^5 - 120*x^6 - x^7)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(-3 + 5*n + n^2)*(-2 + 5*n + n^2)*(5 + 5*n + n^2)/95040.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^7.

A254641 Third partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 131, 2577, 23723, 141694, 636426, 2331462, 7323954, 20396871, 51550213, 120271151, 262391493, 540659756, 1060489444, 1992739932, 3605846676, 6310148349, 10717864983, 17722868317, 28605158351, 45165823626, 69899222030, 106210179010, 158685165990
Offset: 1

Views

Author

Luciano Ancora, Feb 05 2015

Keywords

Links

Crossrefs

Cf. A254640, A254642, A254643.

Programs

  • GAP
    List([1..30], n-> Binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30); # G. C. Greubel, Aug 28 2019
  • Magma
    [Binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30: n in [1..30]]; // G. C. Greubel, Aug 28 2019
  • Maple
    seq(binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30, n=1..30); # G. C. Greubel, Aug 28 2019
  • Mathematica
    Table[n(1+n)(2+n)(3+n)(6 -6n -20n^2 +15n^3 +25n^4 +9n^5 +n^6)/720, {n, 30}]
CoefficientList[Series[(1 +120x +1191x^2 +2416x^3 +1191x^4 +120x^5 + x^6)/(1-x)^11, {x, 0, 30}], x]
Nest[Accumulate,Range[30]^7,3] (* or *) LinearRecurrence[{11,-55,165, -330,462,-462,330,-165,55,-11,1},{1,131,2577,23723, 141694, 636426, 2331462, 7323954,20396871,51550213,120271151},30] (* Harvey P. Dale, Jun 19 2018 *)
  • PARI
    Vec((1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11 + O(x^40)) \\ Andrew Howroyd, Nov 06 2018
  • PARI
    vector(30, n, binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30) \\ G. C. Greubel, Aug 28 2019
  • Sage
    [binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30 for n in (1..30)] # G. C. Greubel, Aug 28 2019

Formula

G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11.
a(n) = n*(1+n)*(2+n)*(3+n)*(6 -6*n -20*n^2 +15*n^3 +25*n^4 +9*n^5 +n^6)/720.
E.g.f.: x (720 +46440*x +262440*x^2 +425910*x^3 +285264*x^4 +92526*x^5 +15600*x^6 +1380*x^7 +60*x^8 +x^9)*exp(x)/6!. - G. C. Greubel, Aug 28 2019

A254869 Seventh partial sums of cubes (A000578).

Original entry on oeis.org

1, 15, 111, 561, 2211, 7293, 21021, 54483, 129558, 286858, 598026, 1184118, 2242266, 4083366, 7184166, 12257850, 20348031, 32951985, 52179985, 80958735, 123288165, 184562235, 271965915, 394962165, 565884540, 800652996, 1119632580, 1548656956
Offset: 1

Views

Author

Luciano Ancora, Feb 17 2015

Keywords

Examples

			2nd differences:   0,  6,  12,  18,   24,   30, ... (A008588)
1st differences:   1,  7,  19,  37,   61,   91, ... (A003215)
-------------------------------------------------------------------
The cubes:         1,  8,  27,  64,  125,  216, ... (A000578)
-------------------------------------------------------------------
1st partial sums:  1,  9,  36, 100,  225,  441, ... (A000537)
2nd partial sums:  1, 10,  46, 146,  371,  812, ... (A024166)
3rd partial sums:  1, 11,  57, 203,  574, 1386, ... (A101094)
4th partial sums:  1, 12,  69, 272,  846, 2232, ... (A101097)
5th partial sums:  1, 13,  82, 354, 1200, 3432, ... (A101102)
6th partial sums:  1, 14,  96, 450, 1650, 5082, ... (A254469)
7th partial sums:  1, 15, 111, 561, 2211, 7293, ... (this sequence)

Links

Crossrefs

Cf. A000537, A000578, A003215, A024166, A101094, A101097, A101102, A254469, A254870, A254871, A254872.

Programs

  • Magma
    [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+7*n+n^2)/604800: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 7 n + n^2)/604800, {n, 26}] (* or *)
CoefficientList[Series[(- 1 - 4 x - x^2)/(- 1 + x)^11, {x, 0, 25}], x]
Nest[Accumulate,Range[30]^3,7] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,15,111,561,2211,7293,21021,54483,129558,286858,598026},30] (* Harvey P. Dale, Apr 24 2017 *)
  • PARI
    vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800) \\ Derek Orr, Feb 19 2015

Formula

G.f.: x*(1 + 4*x + x^2)/(1 - x)^11.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^3.
Sum_{n>=1} 1/a(n) = 1920*sqrt(3/7)*Pi*tan(sqrt(21)*Pi/2) - 251488/49. - Amiram Eldar, Jan 26 2022

A254870 Seventh partial sums of fourth powers (A000583).

Original entry on oeis.org

1, 23, 221, 1355, 6239, 23465, 75803, 217373, 566150, 1361802, 3063502, 6508450, 13159666, 25481470, 47493274, 85567222, 149553199, 254336185, 421956275, 684451365, 1087616985, 1695917535, 2598828765, 3918943275, 5822229660, 8530902276, 12339433068
Offset: 1

Views

Author

Luciano Ancora, Feb 17 2015

Keywords

Examples

			Second differences:   2, 14,  50,  110,  194,   302, ...   A120328(2k+1)
First differences:    1, 15,  65,  175,  369,   671, ...   A005917
--------------------------------------------------------------------------
The fourth powers:    1, 16,  81,  256,  625,  1296, ...   A000583
--------------------------------------------------------------------------
First partial sums:   1, 17,  98,  354,  979,  2275, ...   A000538
Second partial sums:  1, 18, 116,  470, 1449,  3724, ...   A101089
Third partial sums:   1, 19, 135,  605, 2054,  5778, ...   A101090
Fourth partial sums:  1, 20, 155,  760, 2814,  8592, ...   A101091
Fifth partial sums:   1, 21, 176,  936, 3750, 12342, ...   A254681
Sixth partial sums:   1, 22, 198, 1134, 4884, 17226, ...   A254470
Seventh partial sums: 1, 23, 221, 1355, 6239, 23465, ...   (this sequence)

Links

Crossrefs

Cf. A000538, A000583, A005917, A101089, A101090, A101091, A254681, A254470, A254869, A254871, A254872.

Programs

  • Magma
    [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+2*n)*(7 +42*n+6*n^2)/19958400: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 2 n)((7 + 42 n + 6 n^2)/19958400), {n, 24}] (* or *)
CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(- 1 + x)^12, {x, 0, 23}], x]
  • PARI
    vector(50,n,n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400) \\ Derek Orr, Feb 19 2015

Formula

G.f.: (x + 11*x^2 + 11*x^3 + x^4)/(- 1 + x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^4.

A254871 Seventh partial sums of fifth powers (A000584).

Original entry on oeis.org

1, 39, 495, 3705, 19995, 85917, 311493, 989235, 2823990, 7383610, 17931498, 40889862, 88304970, 181852230, 359140470, 683363994, 1257722271, 2246496825, 3905261425, 6623425575, 10983195405, 17840105595, 28431558675, 44521334325, 68589834300, 104081944356
Offset: 1

Views

Author

Luciano Ancora, Feb 17 2015

Keywords

Examples

			Second differences:      30, 180,  570,  1320,  2550, ...   (A068236)
First differences:    1, 31, 211,  781,  2101,  4651, ...   (A022521)
------------------------------------------------------------------------
The fifth powers:     1, 32, 243, 1024,  3125,  7776, ...   (A000584)
------------------------------------------------------------------------
First partial sums:   1, 33, 276, 1300,  4425, 12201, ...   (A000539)
Second partial sums:  1, 34, 310, 1610,  6035, 18236, ...   (A101092)
Third partial sums:   1, 35, 345, 1955,  7990, 26226, ...   (A101099)
Fourth partial sums:  1, 36, 381, 2336, 10326, 36552, ...   (A254644)
Fifth partial sums:   1, 37, 418, 2754, 13080, 49632, ...   (A254682)
Sixth partial sums:   1, 38, 456, 3210, 16290, 65922, ...   (A254471)
Seventh partial sums: 1, 39, 495, 3705, 19995, 85917, ... (this sequence)

Links

Crossrefs

Cf. A000539, A000584, A022521, A101092, A101099, A254471, A254644, A254682, A254869, A254870, A254872.

Programs

  • Magma
    [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(-21+49*n +56*n^2+14*n^3+n^4)/3991680: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015
  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) ((-21 + 49 n + 56 n^2 + 14 n^3 + n^4)/3991680), {n, 23}] (* or *)
CoefficientList[Series[(- 1 - 26 x - 66 x^2 - 26 x^3 - x^4)/(- 1 + x)^13, {x, 0, 22}], x]
  • PARI
    vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(-21 + 49*n + 56*n^2 + 14*n^3 + n^4)/3991680) \\ Derek Orr, Feb 19 2015

Formula

G.f.: (- x - 26*x^2 - 66*x^3 - 26*x^4 - x^5)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(-21 + 49*n + 56*n^2 + 14*n^3 + n^4)/3991680.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^5.
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