A026026 - OEIS

%I #27 May 14 2026 00:57:28

%S 1,3,10,35,125,451,1638,5980,21930,80750,298452,1106921,4118725,

%T 15371475,57528750,215867880,811985790,3061229850,11565545100,

%U 43782423750,166051490514,630877833102,2400830868860,9150602070760,34927775872500,133502608167292

%N a(n) = number of (s(0), s(1), ..., s(2*n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2*n-1) = 4. Also a(n) = T(2*n-1,n-1), where T is defined in A026022.

%H Michael De Vlieger, <a href="/A026026/b026026.txt">Table of n, a(n) for n = 2..1668</a>

%H Paul Drube, <a href="https://arxiv.org/abs/2206.01194">Raised k-Dyck paths</a>, arXiv:2206.01194 [math.CO], 2022. See Appendix pp. 14-15.

%F Expansion of x^2*(1+x^2*C^4)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

%F Conjecture: -(n+3)*(11*n-38)*a(n) +2*(35*n^2-86*n-102)*a(n-1) -4*(13*n-30)*(2*n-5)*a(n-2)=0. - _R. J. Mathar_, Jun 22 2013

%F a(n) = binomial(2*n-3, n-2) - binomial(2*n-3, n-6). - _Christian Krause_, May 09 2026

%F E.g.f.: exp(2*x)*((2+8*x-4*x^2)*BesselI(0, 2*x) - (2 +3*x + 7*x^2 - 4*x^3)*F(2, x^2))/x - 5, where F is the regularized confluent hypergeometric function. - _Stefano Spezia_, May 11 2026

%t CoefficientList[Series[x^2*(1 + x^2*#^4)*#^3 &[(1 - (1 - 4*x)^(1/2))/(2*x)], {x, 0, 25}], x] (* _Michael De Vlieger_, Jan 21 2026 *)

%Y Cf. A000108, A026022.

%K nonn,easy

%O 2,2

%A _Clark Kimberling_

%E Definition corrected by _Herbert Kociemba_, May 02 2004