cp's OEIS Frontend

A006419 a(n) = 2^(2*n+1) - C(2*n+3,n+1) + C(2*n+1,n).

%I A006419 M4419 #76 Jul 29 2025 08:38:26
%S A006419 0,1,7,37,176,794,3473,14893,63004,263950,1097790,4540386,18696432,
%T A006419 76717268,313889477,1281220733,5219170052,21224674118,86188320962,
%U A006419 349550141078,1416102710912,5731427140268,23177285611082,93655986978002,378195990166136,1526289367335244
%N A006419 a(n) = 2^(2*n+1) - C(2*n+3,n+1) + C(2*n+1,n).
%C A006419 Number of rooted isthmusless planar maps with n+1 faces and 2 vertices. - _Dan Drake_, Aug 08 2005
%C A006419 a(n) = total area below all Dyck (n+1)-paths and above the lowest possible Dyck path, namely, UDUD...UD (taking upsteps of unit length). For example, the areas below the 5 Dyck 3-paths UUUDDD, UUDUDD, UDUUDD, UUDDUD, UDUDUD are 3,2,1,1,0 respectively, yielding a(2)=3+2+1+1+0=7. - _David Callan_, Jul 03 2006
%C A006419 Convolution of A000245 and A000302 (powers of 4).- _Philippe Deléham_, Jun 02 2013
%D A006419 D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
%D A006419 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006419 Seiichi Manyama, <a href="/A006419/b006419.txt">Table of n, a(n) for n = 0..1660</a>
%H A006419 Jason Bandlow and Kendra Killpatrick, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v8i1r40">An area-to-inv bijection between Dyck paths and 312-avoiding permutations</a>,Electron. J. Combin. 8 (2001), no. 1, Research Paper 40, 16 pp.
%H A006419 Miklós Bóna, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i1p62">Surprising Symmetries in Objects Counted by Catalan Numbers</a>, Electronic J. Combin., 19 (2012), #P62, eq. (5).
%H A006419 Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa <a href="https://doi.org/10.1155/2014/316535">Noncontiguous pattern containment in binary trees</a>, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Table 1.
%H A006419 R. P. Stanley, F. Zanello, <a href="http://arxiv.org/abs/1312.4352">The Catalan case of Armstrong's conjecture on core partitions</a>, arXiv preprint arXiv:1312.4352 [math.CO], 2013.
%H A006419 T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259.
%F A006419 a(n+1) = Sum_{k=0..n} (n-k)*A000108(n-k)*A001700(k). - _Philippe Deléham_, Jan 25 2004
%F A006419 G.f.: c(x)^3*x/(1-4x) where c(x) = g.f. for the Catalan numbers A000108. - _Philippe Deléham_, Jun 02 2013
%F A006419 a(n) = Integral_{x=0..4} x^n*W(x)*dx, n >= 0, is the integral representation as n-th moment of a signed weight function W(x), where W(x) = W_a(x) + W_c(x), with W_a(x) = 2*Dirac(x-4), which is the discrete (atomic) part, and W_c(x) = (1/(2*Pi))*(1-x)*sqrt(x/(4-x)) is the continuous part of W(x): W_c(0) = W_c(1) = 0, W_c(x) > 0 for x < 1, lim_{x->4} W_c(x) = -oo. - _Karol A. Penson_, Jul 31 2013 [edited by _Michel Marcus_, Mar 14 2020]
%F A006419 (n+2)*a(n) + (-9*n-10)*a(n-1) + 2*(12*n+1)*a(n-2) + 8*(-2*n+3)*a(n-3) = 0. - _R. J. Mathar_, Mar 30 2014
%F A006419 a(n) = Sum_{k=0..n} binomial(2*(n+1), n-k-1). - _Vladimir Kruchinin_, Oct 23 2016
%F A006419 0 = a(n)*(+256*a(n+1) - 992*a(n+2) + 520*a(n+3) - 72*a(n+4)) + a(n+1)*(+224*a(n+1) + 344*a(n+2) - 398*a(n+3) + 70*a(n+4)) + a(n+2)*(+6*a(n+2) + 59*a(n+3) - 17*a(n+4)) + a(n+3)*(-a(n+3) + a(n+4)), for all n >= 0. - _Michael Somos_, Oct 23 2016
%F A006419 a(n) = [x^n] x/((1 - 2*x)*(1 - x)^(n+3)). - _Ilya Gutkovskiy_, Oct 25 2017
%F A006419 From _Seiichi Manyama_, Jul 29 2025: (Start)
%F A006419 a(n) = Sum_{k=0..n-1} binomial(2*k+1+l,k) * binomial(2*n-2*k-l,n-k-1) for every real number l.
%F A006419 a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(n+k+2,k). (End)
%e A006419 G.f. = x + 7*x^2 + 37*x^3 + 176*x^4 + 794*x^5 + 3473*x^6 + 14893*x^7 + 63004*x^8 + ...
%p A006419 f := n->2^(2*n+1)-binomial(2*n+3,n+1)+binomial(2*n+1,n); seq(f(n), n=0..30);
%t A006419 Table[2^(2 n + 1) - Binomial[2 n + 3, n + 1] +
%t A006419 Binomial[2 n + 1, n], {n, 0, 30}] (* _Wesley Ivan Hurt_, Mar 30 2014 *)
%o A006419 (Maxima)
%o A006419 a(n):=sum(binomial(2*(n+1),n-k-1),k,0,n); /* _Vladimir Kruchinin_, Oct 23 2016 */
%Y A006419 A diagonal of A342981.
%Y A006419 Cf. A386612, A386614, A386616, A386617.
%K A006419 nonn
%O A006419 0,3
%A A006419 _N. J. A. Sloane_