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 11-16 of 16 results.

A000967 Sum of Fermat coefficients.

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 91, 210, 492, 1165, 2786, 6710, 16267, 39650, 97108, 238824, 589521, 1459960, 3626213, 9030450, 22542396, 56393792, 141358274, 354975429, 892893120, 2249412290, 5674891000, 14335757256, 36259245522, 91815545800
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			n...Sum_{c=1..n} (n:c).....a(n)
--------------------------------
.1........1.................1
.2........2.................2
.3........4.................4
.4........8+1/3.............8

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A143858, A258708.

Programs

  • Haskell
    import Data.Function (on)
a000967 n = round $ sum $
            zipWith ((/) `on` fromIntegral) (a258993_row n) [1, 3 ..]
-- Reinhard Zumkeller, Jun 22 2015
  • Magma
    [Round((&+[Binomial(n+k,n-k)/(2*k+1): k in [0..n-1]])): n in [1..35]]; // G. C. Greubel, Apr 16 2019
  • Maple
    FermatCoeff:=(n,c)->binomial(2*n-c,c-1)/c:seq(round(add(FermatCoeff(n,c),c=1..n)),n=1..40); # Pab Ter, Oct 13 2005
  • Mathematica
    Table[Round[Sum[Binomial[n+k, n-k]/(2*k+1), {k, 0, n-1}]], {n,1,35}] (* G. C. Greubel, Apr 16 2019 *)
  • PARI
    {a(n) = round(sum(k=0,n-1, binomial(n+k,n-k)/(2*k+1)))}; \\ G. C. Greubel, Apr 16 2019
  • Sage
    [round(sum(binomial(n+k,n-k)/(2*k+1) for k in (0..n-1))) for n in (1..35)] # G. C. Greubel, Apr 16 2019

Formula

Following Piza's definition for the Fermat coefficients: (n:c)=binomial(2n-c, c-1)/c, a(n)= Round( sum_ {c=1..n} (n:c) ). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005
a(n) = rounded(sum(A258993(n,k)/(2*k+1)): k = 0..n-1). - Reinhard Zumkeller, Jun 22 2015

Extensions

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005

A000970 Fermat coefficients.

Original entry on oeis.org

1, 7, 25, 66, 143, 273, 476, 775, 1197, 1771, 2530, 3510, 4750, 6293, 8184, 10472, 13209, 16450, 20254, 24682, 29799, 35673, 42375, 49980, 58565, 68211, 79002, 91025, 104371, 119133, 135408, 153296, 172900, 194327, 217686, 243090, 270655
Offset: 5

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

Crossrefs

Cf. A258708.

Programs

  • Haskell
    a000970 n = a258708 n (n - 5)  -- Reinhard Zumkeller, Jun 23 2015
  • Maple
    A000970:=-(2*z**4+3*z**5+3*z**2+4*z**3+3*z+1)/(z**4+z**3+z**2+z+1)/(z-1)**5; # Simon Plouffe in his 1992 dissertation
  • Mathematica
    CoefficientList[Series[(3x^5+2x^4+4x^3+3x^2+3x+1)/((1-x^5)(1-x)^4),{x,0,50}],x] (* Vincenzo Librandi, Mar 28 2012 *)
LinearRecurrence[{4,-6,4,-1,1,-4,6,-4,1},{1,7,25,66,143,273,476,775,1197},40] (* Harvey P. Dale, Sep 06 2017 *)
  • PARI
    Vec((3*x^5+2*x^4+4*x^3+3*x^2+3*x+1)/(1-x^5)/(1-x)^4+O(x^99)) \\ Charles R Greathouse IV, Mar 28 2012

Formula

G.f.: x^5(3x^5 + 2x^4 + 4x^3 + 3x^2 + 3x + 1)/((1-x^5)(1-x)^4).
a(n) = A258708(n,n-5) = A258708(2*n-7,2). - Reinhard Zumkeller, Jun 23 2015

Extensions

More terms from Sean A. Irvine, Sep 25 2011

A000971 Fermat coefficients.

Original entry on oeis.org

1, 9, 42, 132, 334, 728, 1428, 2584, 4389, 7084, 10963, 16380, 23751, 33563, 46376, 62832, 83657, 109668, 141778, 181001, 228459, 285384, 353127, 433160, 527085, 636636, 763686, 910252, 1078500, 1270752, 1489488, 1737355, 2017169, 2331924
Offset: 6

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A258708.

Programs

  • Haskell
    a000971 n = a258708 n (n - 6)  -- Reinhard Zumkeller, Jun 23 2015
  • Maple
    (1+3*z+3*z^7+z^8+3*z^2-4*z^3+10*z^4-4*z^5+3*z^6)/(z^6+z^3+1)/(-1+z)^6;
  • Mathematica
    CoefficientList[Series[(1+3*x+3*x^7+x^8+3*x^2-4*x^3+10*x^4-4*x^5+3*x^6)/(x^6+x^3+1)/(-1+x)^6,{x,0,40}],x] (* Vincenzo Librandi, Mar 28 2012 *)
  • PARI
    Vec((1+3*z+3*z^7+z^8+3*z^2-4*z^3+10*z^4-4*z^5+3*z^6)/(z^6+z^3+1)/(z-1)^6+O(x^99)) \\ Charles R Greathouse IV, Mar 28 2012

Formula

G.f.: (1 + 3x + 3x^7 + x^8 + 3x^2 - 4x^3 + 10x^4 - 4x^5 + 3x^6)/(x^6 + x^3 + 1)/(-1+x)^6 (see MAPLE line).
a(n) = A258708(n,n-6). - Reinhard Zumkeller, Jun 23 2015

Extensions

More terms from Sean A. Irvine, Sep 25 2011

A000972 Fermat coefficients.

Original entry on oeis.org

1, 12, 66, 245, 715, 1768, 3876, 7752, 14421, 25300, 42287, 67860, 105183, 158224, 231880, 332112, 466089, 642341, 870922, 1163580, 1533939, 1997688, 2572780, 3279640, 4141382, 5184036, 6436782, 7932196, 9706503, 11799840, 14256528, 17125353, 20459857
Offset: 7

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A258708.

Programs

  • Haskell
    a000972 n = a258708 n (n - 7)  -- Reinhard Zumkeller, Jun 23 2015
  • Maple
    a := n->floor((2*n)*(2*n+1)*(2*n+2)*(2*n+3)*(2*n+4)*(2*n+5)/7!);
  • Mathematica
    Table[Floor[((2*n)*(2*n+1)*(2*n+2)*(2*n+3)*(2*n+4)*(2*n+5)/7!)],{n,1,30}] (* Vincenzo Librandi, Apr 10 2012 *)
With[{c=7!,t=Times@@(2n+Range[0,5])},Table[Floor[t/c],{n,30}]] (* Harvey P. Dale, Apr 20 2014 *)
  • PARI
    Vec(x^7*(1 + 6*x + 9*x^2 + 9*x^3 + 10*x^4 + 7*x^5 + 12*x^6 + 6*x^7 + 4*x^8) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^50)) \\ Colin Barker, Mar 28 2017

Formula

a(n) = A258708(n,n-7). - Reinhard Zumkeller, Jun 23 2015
G.f.: x^7*(1 + 6*x + 9*x^2 + 9*x^3 + 10*x^4 + 7*x^5 + 12*x^6 + 6*x^7 + 4*x^8) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Colin Barker, Mar 28 2017

A011793 Triangle of numbers of irreducible Euler sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 3, 1, 2, 5, 3, 1, 1, 5, 7, 4, 1, 3, 8, 9, 4, 1, 1, 7, 14, 12, 5, 1, 3, 14, 20, 15, 5, 1, 1, 9, 25, 30, 18, 6, 1, 4, 20, 42, 40, 22, 6, 1, 1, 12, 42, 66, 55, 26, 7, 1
Offset: 1

Views

Author

N. J. A. Sloane, David Broadhurst

Keywords

Links

Crossrefs

Cf. A258708.

Programs

  • Mathematica
    t[n_, k_] := (2/(n+k))*Sum[ If[ EvenQ[d], MoebiusMu[d/2]*Binomial[(n+k)/d, (n-k)/d], 0], {d, Intersection[ Divisors[n+k], Divisors[n-k]]}]; t[1, 1] = t[2, 1] = 1;row[1] = row[2] = {1}; row[n_] := Table[t[n, k], {k, 2 - Mod[n, 2], n-1, 2}]; Flatten[ Table[ row[n], {n, 1, 17}]] (* Jean-François Alcover, Jun 15 2012, after David Broadhurst *)

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 1, 14, 49, 49, 1, 30, 243, 729, 729, 1, 55, 847, 5324, 14641, 14641, 1, 91, 2366, 26364, 142805, 371293, 371293, 1, 140, 5670, 101250, 928125, 4556250, 11390625, 11390625, 1, 204, 12138, 324258, 4593655, 36916282, 168962983, 410338673, 410338673
Offset: 0

Views

Author

Peter Bala, Mar 12 2024

Keywords

Comments

The table entries are integers since a(n, k) := binomial(n+k, n-k)/(2*k + 1) * (2*n + 1) gives the entries of the transpose of triangle A082985.

Examples

			Triangle begins
 n\k | 0    1      2       3        4        5        6
 - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   0 | 1
   1 | 1    1
   2 | 1    5      5
   3 | 1   14     49      49
   4 | 1   30    243     729      729
   5 | 1   55    847    5324    14641    14641
   6 | 1   91   2366   26364   142805   371293   371293
  ...

Crossrefs

A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).
Cf. A008310, A082985, A258708, A370259, A370260.

Programs

  • Maple
    seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);
  • Mathematica
    Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)

Formula

n-th row polynomial R(n, x) = Sum_{k = 0..n} T(n, k)*x^k = sqrt( 2* Sum_{k = 0..2*n} (2*n + 1)^(k-1) *binomial(2*n+k+2, 2*k+2)/(2*n + k + 2) * x^k ).
R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).
R(n, 2) = A370260(n).
Previous Showing 11-16 of 16 results.

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

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