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.
%I A137644 #43 Feb 17 2024 05:19:27
%S A137644 1,3,16,95,591,3780,24620,162423,1081780,7258053,48982176,332140328,
%T A137644 2261099491,15444137880,105789736896,726426836103,4998885106599,
%U A137644 34464824536500,238017084356680,1646234203000485,11401464090042224,79060352485691272,548829398923188036
%N A137644 a(n) = Sum_{k=0..n} C(n+k,k)*C(n+k,n-k).
%C A137644 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (0,1), (0,2). - _Eric Werley_, Jun 29 2011
%C A137644 Diagonal of rational function 1/(1 - (x + y + x*y + x^2)). - _Gheorghe Coserea_, Aug 31 2018
%H A137644 Indranil Ghosh, <a href="/A137644/b137644.txt">Table of n, a(n) for n = 0..500</a>
%F A137644 a(n) = 3F2( {-n, n+1, n+1}; {1/2, 1})( -(1/4) ). - _Olivier Gérard_, Apr 23 2009
%F A137644 G.f.: F'(x)/(1+F(x)), where F(x)=x*(1+F(x))/(1-F(x)-F(x)^2). - _Vladimir Kruchinin_, Mar 24 2012
%F A137644 a(n) = A063967(n,n). - _Alois P. Heinz_, Oct 11 2017
%F A137644 a(n) ~ sqrt(56 + (7*(15953 - 267*sqrt(105)))^(1/3) + (7*(15953 + 267*sqrt(105)))^(1/3)) * (((36 + (44766 - 1050*sqrt(105))^(1/3) + (6*(7461 + 175*sqrt(105)))^(1/3))/15)^n / sqrt(210*Pi*n)). - _Vaclav Kotesovec_, Feb 17 2024
%e A137644 The triangle of number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (0,1), (0,2) begins:
%e A137644 1;
%e A137644 1, 3;
%e A137644 1, 5, 16;
%e A137644 1, 7, 29, 95;
%e A137644 1, 9, 46, 179, 591;
%e A137644 1, 11, 67, 303, 1140, 3780;
%e A137644 1, 13, 92, 475, 2010, 7405, 24620;
%e A137644 1, 15, 121, 703, 3309, 13427, 48761, 162423;
%e A137644 1, 17, 154, 995, 5161, 22892, 90241, 324317, 1081780;
%e A137644 This sequence is the diagonal. - _Joerg Arndt_, Jul 01 2011
%t A137644 Table[ HypergeometricPFQ[{-n, 1 + n, 1 + n}, {1/2, 1}, -(1/4)], {n,0,20}] (* _Olivier Gérard_, Apr 23 2009 *)
%t A137644 Table[Sum[Binomial[n+k,k]Binomial[n+k,n-k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Aug 03 2011 *)
%o A137644 (PARI) a(n)=sum(k=0,n,binomial(n+k,k)*binomial(n+k,n-k))
%o A137644 (PARI) /* same as in A092566 but use */
%o A137644 steps=[[1,0], [1,1], [0,1], [0,2]];
%o A137644 /* _Joerg Arndt_, Jun 30 2011 */
%Y A137644 Cf. A063967.
%K A137644 nonn
%O A137644 0,2
%A A137644 _Paul D. Hanna_, Jan 31 2008