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.

Showing 1-5 of 5 results.

A089658 a(n) = S1(n,1), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 11, 42, 136, 400, 1104, 2912, 7424, 18432, 44800, 107008, 251904, 585728, 1347584, 3072000, 6946816, 15597568, 34799616, 77201408, 170393600, 374341632, 818937856, 1784676352, 3875536896, 8388608000, 18102616064, 38956695552, 83617644544, 179046449152
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S1(n, t): A001792 (t=0), this sequence (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

Programs

  • Magma
    I:=[0,2,11]; [n le 3 select I[n] else 6*Self(n-1)-12*Self(n-2)+8*Self(n-3): n in [1..41]]; // Vincenzo Librandi, Jun 22 2016
  • Mathematica
    LinearRecurrence[{6,-12,8}, {0,2,11}, 40] (* Vincenzo Librandi, Jun 22 2016 *)
  • SageMath
    [n*(5+3*n)*2^(n-3) for n in (0..40)] # G. C. Greubel, May 24 2022

Formula

a(n) = n*(5 + 3*n) * 2^(n-3). (See Wang and Zhang p. 333.)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 2.
G.f.: x*(2 - x)/(1 - 2*x)^3. (End)
E.g.f.: x*(4 + 3*x)*exp(2*x)/2. - Ilya Gutkovskiy, Jun 21 2016
a(n) = 2*A001788(n) - A001788(n-1). - R. J. Mathar, Jul 22 2021

A089659 a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 19, 104, 440, 1600, 5264, 16128, 46848, 130560, 352000, 923648, 2369536, 5963776, 14766080, 36044800, 86900736, 207224832, 489357312, 1145569280, 2660761600, 6136266752, 14060355584, 32027705344, 72561459200, 163577856000, 367068708864, 820204535808
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Columns : A000012, A000012, A000984, A006480, A008977, A008978.
Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), this sequence (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

Programs

  • Magma
    I:=[0,2,19,104]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..41]]; // Vincenzo Librandi, Jun 22 2016
  • Mathematica
    LinearRecurrence[{8,-24,32,-16}, {0,2,19,104}, 40] (* Vincenzo Librandi, Jun 22 2016 *)
  • SageMath
    [2^(n-3)*n*(7*n^2 + 12*n + 5)/3 for n in (0..40)] # G. C. Greubel, May 24 2022

Formula

a(n) = 2^(n-3)*n*(7*n^2 + 12*n + 5)/3. (see Wang and Zhang p. 333)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 3.
G.f.: x*(2 + 3*x)/(1 - 2*x)^4. (End)
E.g.f.: x*(12 + 33*x + 14*x^2)*exp(2*x)/6. - Ilya Gutkovskiy, Jun 21 2016

A089660 a(n) = S1(n,3), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 35, 276, 1522, 6820, 26664, 94640, 312512, 975744, 2913280, 8386048, 23416320, 63724544, 169637888, 443043840, 1137934336, 2879979520, 7194083328, 17761304576, 43390730240, 104997322752, 251881062400, 599482433536, 1416470986752, 3324615065600
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), A089659 (t=2), this sequence (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

Programs

  • Magma
    [2^(n-6)*n*(3*n*(7+10*n+5*n^2) -2): n in [0..40]]; // G. C. Greubel, May 24 2022
  • Mathematica
    LinearRecurrence[{10,-40,80,-80,32}, {0,2,35,276,1522}, 40] (* Vincenzo Librandi, Jun 22 2016 *)
  • SageMath
    [n*(15*n^3+30*n^2+21*n-2)*2^(n-6) for n in (0..40)] # G. C. Greubel, May 24 2022

Formula

a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*2^(n-6). - R. J. Mathar, Sep 16 2009
G.f.: x*(2 + 15*x + 6*x^2 + 2*x^3)/(1-2*x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5) for n > 4. - Chai Wah Wu, Jun 21 2016
E.g.f.: x*(8 + 54*x + 60*x^2 + 15*x^3)*exp(2*x)/4. - Ilya Gutkovskiy, Jun 21 2016

A089661 a(n) = S1(n,4), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

Original entry on oeis.org

0, 2, 67, 764, 5492, 30304, 140672, 577920, 2167680, 7577088, 25037056, 79016960, 240028672, 705961984, 2019713024, 5641535488, 15431565312, 41438281728, 109462880256, 284942925824, 732004876288, 1858158460928, 4665915736064, 11600782163968, 28582042664960
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S1(n,t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), this sequence (t=4), A089662 (t=5), A089663 (t=6).

Programs

  • Magma
    [2^(n-5)*n*(93*n^4+225*n^3+185*n^2+15*n-38)/15: n in [0..30]]; // Vincenzo Librandi, Jun 22 2016
  • Mathematica
    LinearRecurrence[{12,-60,160,-240,192,-64}, {0,2,67,764,5492,30304}, 40] (* Vincenzo Librandi, Jun 22 2016 *)
  • SageMath
    [n*(n+1)*(93*n^3 +132*n^2 +53*n -38)*2^(n-5)/15 for n in (0..40)] # G. C. Greubel, May 24 2022

Formula

a(n) = (1/15)*n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*2^(n-5). (See Wang and Zhang, p. 334)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 12*a(n-1) - 60*a(n-2) + 160*a(n-3) - 240*a(n-4) + 192*a(n-5) - 64*a(n-6) for n > 5.
G.f.: x*(2 + 43*x + 80*x^2 + 24*x^3)/(1 - 2*x)^6. (End)
a(n) = 2^(n-5)*n*(93*n^4 + 225*n^3 + 185*n^2 + 15*n - 38)/15. - Ilya Gutkovskiy, Jun 21 2016
E.g.f.: (1/30)*x*(60 + 885*x + 1930*x^2 + 1155*x^3 + 186*x^4)*exp(2*x). - G. C. Greubel, May 24 2022

A089662 a(n) = S1(n,5), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

Original entry on oeis.org

0, 2, 131, 2172, 20386, 138580, 763824, 3631712, 15470144, 60527232, 221297920, 765580288, 2529498624, 8039103488, 24713744384, 73818562560, 215011065856, 612515381248, 1710842904576, 4695105732608, 12682107944960, 33768108982272, 88748191645696
Offset: 0

Views

Author

N. J. A. Sloane, Jan 04 2004

Keywords

Links

Crossrefs

Sequences of S1(n,t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), this sequence (t=5), A089663 (t=6).

Programs

  • Magma
    [2^(n-7)*n*(21*n^5+61*n^4+55*n^3+15*n^2-28*n+4): n in [0..30]]; // Vincenzo Librandi, Jun 22 2016
  • Mathematica
    LinearRecurrence[{14,-84,280,-560,672,-448,128}, {0,2,131,2172,20386,138580, 763824}, 30] (* Vincenzo Librandi, Jun 22 2016 *)
  • SageMath
    [2^(n-7)*n*(21*n^5 +61*n^4 +55*n^3 +15*n^2 -28*n +4) for n in (0..40)] # G. C. Greubel, May 24 2022

Formula

There is an explicit formula for the sum - see Wang and Zhang, p. 334.
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 14*a(n-1) - 84*a(n-2) + 280*a(n-3) - 560*a(n-4) + 672*a(n-5) - 448*a(n-6) + 128*a(n-7) for n > 6.
G.f.: x*(-16*x^5 + 64*x^4 + 422*x^3 + 506*x^2 + 103*x + 2)/(1 - 2*x)^7. (End)
a(n) = 2^(n-7)*n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4). - Ilya Gutkovskiy, Jun 21 2016
E.g.f.: (1/4)*x*(8 + 246*x + 940*x^2 + 1015*x^3 + 376*x^4 + 42*x^5)*exp(2*x). - G. C. Greubel, May 24 2022
Showing 1-5 of 5 results.

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

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