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.

Previous Showing 11-14 of 14 results.

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.

A254872 Seventh partial sums of sixth powers (A001014).

Original entry on oeis.org

1, 71, 1205, 11075, 70295, 345857, 1409387, 4962365, 15539750, 44192010, 115917118, 283828498, 654885730, 1434717550, 3002927770, 6035661334, 11699568079, 21951176425, 39988722875, 70920437325, 122735050305
Offset: 1

Views

Author

Luciano Ancora, Feb 17 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, ... (A254472)
Seventh partial sums: 1, 71, 1205, 11075, 70295, ... (this sequence)

Links

Crossrefs

Cf. A000540, A001014, A022522, A101093, A254472, A254640, A254645, A254683, A254869, A254870, A254871.

Programs

  • Mathematica
    Table[(n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 2 n) (- 49 + 147 n^2 + 42 n^3 + 3 n^4))/51891840, {n, 21}] (* or *)
CoefficientList[Series[(1 + 57 x + 302 x^2 + 302 x^3 + 57 x^4 + x^5)/(- 1 + x)^14, {x, 0, 20}], x]

Formula

G.f.: (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)/(- 1 + x)^14.
a(n) = (n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(- 49 + 147*n^2 + 42*n^3 + 3*n^4))/51891840.
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^6.

A255178 Second differences of eighth powers (A001016).

Original entry on oeis.org

1, 254, 6050, 52670, 266114, 963902, 2796194, 6927230, 15257090, 30683774, 57405602, 101263934, 170126210, 274309310, 427043234, 644975102, 948713474, 1363412990, 1919399330, 2652834494, 3606422402, 4830154814, 6382097570, 8329217150, 10748247554
Offset: 0

Views

Author

Luciano Ancora, Feb 21 2015

Keywords

Examples

			Second differences:  1, 254, 6050, 52670, 266114, ... (this sequence)
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, ... (A254642)
Fourth partial sums: 1, 260, 7595, 94360, 723534, ... (A254647)

Links

Crossrefs

Cf. A000542, A001016, A022524, A253636, A254642, A254647, A255177.

Programs

  • Magma
    [n eq 0 select 1 else 2*(28*n^6+70*n^4+28*n^2+1): n in [0..30]]; // Vincenzo Librandi, Mar 12 2015
  • Mathematica
    Join[{1}, Table[2 (28 n^6 + 70 n^4 + 28 n^2 + 1), {n, 1, 30}]]
Join[{1},Differences[Range[0,30]^8,2]] (* Harvey P. Dale, Aug 26 2024 *)

Formula

G.f.: (1 + x)*(1 + 246*x + 4047*x^2 + 11572*x^3 + 4047*x^4 + 246*x^5 + x^6)/(1 - x)^7.
a(n) = 2*(28*n^6 + 70*n^4 + 28*n^2 + 1) for n>0, a(0)=1.

Extensions

Edited by Bruno Berselli, Mar 19 2015
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.