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 1-4 of 4 results.

A038731 Number of columns in all directed column-convex polyominoes of area n+1.

Original entry on oeis.org

1, 3, 10, 32, 99, 299, 887, 2595, 7508, 21526, 61251, 173173, 486925, 1362627, 3797374, 10543724, 29180067, 80521055, 221610563, 608468451, 1667040776, 4558234018, 12441155715, 33900136297, 92230468249, 250570010499, 679844574322, 1842280003640
Offset: 0

Views

Author

Clark Kimberling, May 02 2000

Keywords

Comments

Apply Riordan array (1/(1-x), x/(1-x)^2) to n+1. - Paul Barry, Oct 13 2009
Binomial transform of (A001629 shifted left twice). - R. J. Mathar, Feb 06 2010

Links

Crossrefs

Row-sums of array T as in A038730.
First differences of A030267.
Row sums of A318942(n+1).
Cf. A000045.

Programs

  • Haskell
    a038731 n = a038731_list !! n
a038731_list = c [1] $ tail a000045_list where
   c us vs'@(v:vs) = (sum $ zipWith (*) us vs') : c (v:us) vs
-- Reinhard Zumkeller, Oct 31 2013
  • Magma
    I:=[1, 3, 10, 32]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 04 2012
  • Mathematica
    Table[Sum[Binomial[n, k]*CoefficientList[Series[1/(1 - x - x^2)^2, {x, 0, k}], x][[-1]], {k, 0, n}], {n, 0, 27}] (* Arkadiusz Wesolowski, Feb 03 2012 *)
LinearRecurrence[{6, -11, 6, -1}, {1, 3, 10, 32}, 30] (* Vincenzo Librandi, Feb 04 2012 *)

Formula

5*a(n) = (2n+1)*F(2n+2) - (n-4)*F(2n+1), where the F(n)'s are the Fibonacci numbers, F(0)=0, F(1)=1.
a(n) = Sum_{k=1..n+1} k*binomial(n+k-1, 2k-2). - Emeric Deutsch, Jun 11 2003
From Paul Barry, Oct 13 2009: (Start)
G.f.: (1-x)^3/(1-3x+x^2)^2.
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(k+1). (End)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - R. J. Mathar, Feb 06 2010
a(n) = Sum_{k=0..n} (F(2k)+0^k)*F(2n-2k+1). - Paul Barry, Jun 23 2010
E.g.f.: exp(3*x/2)*(5*(5 + 4*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(7 + 10*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

Extensions

Entry improved by comments from Emeric Deutsch, Jun 14 2001

A318941 Number of Dyck paths with n nodes and altitude 2.

Original entry on oeis.org

0, 0, 1, 4, 12, 35, 99, 274, 747, 2015, 5394, 14359, 38067, 100610, 265299, 698359, 1835922, 4821695, 12653739, 33188674, 87010587, 228039695, 597501714, 1565251879, 4099826787, 10737374210, 28118587299, 73630970599, 192799490322, 504817832015
Offset: 0

Views

Author

N. J. A. Sloane, Sep 18 2018

Keywords

Links

Crossrefs

A column of A318942.

Programs

  • Maple
    (1-x)^2*x^2*(1+x)/(1-2*x)/(1-3*x+x^2) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ; # R. J. Mathar, Nov 25 2018
  • PARI
    concat([0,0], Vec(x^2*(1 - x)^2*(1 + x) / ((1 - 2*x)*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Apr 09 2019

Formula

From Colin Barker, Apr 09 2019: (Start)
a(n) = 2^(-3-n)*(-3*4^n + 4*(3-sqrt(5))^n*(3+sqrt(5)) - 4*(-3+sqrt(5))*(3+sqrt(5))^n) for n>2.
a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) n>5.
(End)
Note that Czabarka et al. give a g.f. for the whole triangle. - N. J. A. Sloane, Apr 09 2019
a(n) = A005248(n-1) -3*2^(n-3), n>=3. [Czabarka, Proposition 5 (2)] - R. J. Mathar, Apr 09 2019

A318943 Number of Dyck paths with n nodes and altitude 3.

Original entry on oeis.org

0, 0, 0, 1, 6, 21, 68, 208, 612, 1752, 4916, 13588, 37128, 100548, 270404, 723208, 1925844, 5110644, 13524872, 35713828, 94140900, 247806600, 651572660, 1711695508, 4493475336, 11789439876, 30917835908, 81053196808, 212426303892, 556607396532
Offset: 0

Views

Author

N. J. A. Sloane, Sep 18 2018

Keywords

Links

Crossrefs

A column of A318942.

Programs

  • Maple
    (1-x)^2*x^3*(1+x-3*x^2)/(1-2*x)^2/(1-3*x+x^2) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ; # R. J. Mathar, Nov 25 2018
  • Mathematica
    LinearRecurrence[{7, -17, 16, -4}, {0, 0, 0, 1, 6, 21, 68, 208}, 50] (* Paolo Xausa, May 24 2024 *)
  • PARI
    concat([0,0,0], Vec(x^3*(1 - x)^2*(1 + x - 3*x^2) / ((1 - 2*x)^2*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Apr 11 2019

Formula

a(n) = 8*A001906(n+1)-20*A001906(n)-2^(n-5)*(16+3*n), n>=4. - R. J. Mathar, Apr 09 2019
a(n) = 7*a(n-1) - 17*a(n-2) + 16*a(n-3) - 4*a(n-4) for n>7. - Colin Barker, Apr 11 2019

A318944 Number of Dyck paths with n nodes and altitude 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 32, 114, 376, 1177, 3549, 10406, 29861, 84249, 234502, 645625, 1761765, 4772534, 12851261, 34434561, 91890118, 244385617, 648139821, 1714976054, 4529163125, 11942440233, 31448759302, 82727323369, 217426319541, 571033273142
Offset: 0

Views

Author

N. J. A. Sloane, Sep 18 2018

Keywords

Links

Crossrefs

A column of A318942.

Programs

  • Maple
    (1-x)^2*x^4*(1+x-8*x^2+7*x^3)/(1-2*x)^3/(1-3*x+x^2) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ; # R. J. Mathar, Nov 25 2018
  • Mathematica
    LinearRecurrence[{9,-31,50,-36,8},{0,0,0,0,1,8,32,114,376,1177},30] (* Harvey P. Dale, Nov 03 2019 *)
  • PARI
    concat([0,0,0,0], Vec(x^4*(1 - x)^2*(1 + x - 8*x^2 + 7*x^3) / ((1 - 2*x)^3*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Apr 11 2019

Formula

a(n) = 9*a(n-1) - 31*a(n-2) + 50*a(n-3) - 36*a(n-4) + 8*a(n-5) for n>9. - Colin Barker, Apr 11 2019
Showing 1-4 of 4 results.

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

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