A025273 -id:A025273 - cp's OEIS frontend

A217333 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-x)^k ).

1, 1, 2, 5, 12, 29, 72, 182, 466, 1207, 3158, 8334, 22158, 59299, 159614, 431838, 1173710, 3203244, 8774780, 24118522, 66497316, 183858411, 509670494, 1416231616, 3944027402, 11006186760, 30772507128, 86191006746, 241815195292, 679488418879, 1912123070998

Offset: 0

Paul D. Hanna, Sep 30 2012

nonn

The radius of convergence of g.f. A(x) is r = 0.339332122592393190... where 1-4*r+4*r^2-4*r^3+4*r^4 = 0, with A(r) = (1-2*r)/(2*r^3) = 4.112009743749...

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 29*x^5 + 72*x^6 + 182*x^7 +...

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Cf. A025273, A217282.

(1 - 2x - Sqrt[1 - 4x + 4x^2 - 4x^3 + 4x^4])/(2x^3) + O[x]^31 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 27 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k/(1-x+x*O(x^n))^k))),n)}

{a(n)=polcoeff((1-2*x - sqrt(1-4*x+4*x^2-4*x^3+4*x^4 +x^4*O(x^n)))/(2*x^3),n)}
for(n=0,40,print1(a(n),", "))

G.f.: (1-2*x - sqrt(1-4*x+4*x^2-4*x^3+4*x^4))/(2*x^3).
Conjecture: (n+3)*a(n) +2*(-2*n-3)*a(n-1) +4*n*a(n-2) +2*(-2*n+3)*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, May 17 2019
G.f. A(x) satisfies: A(x) = 1 + x * (1 + x^2*A(x)^2) / (1 - 2*x). - Ilya Gutkovskiy, Jun 30 2020

A307788 Number of valid hook configurations of permutations of [n] that avoid the patterns 231 and 321.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 29, 72, 182, 466, 1207, 3158, 8334, 22158, 59299, 159614, 431838, 1173710, 3203244, 8774780, 24118522, 66497316, 183858411, 509670494, 1416231616, 3944027402, 11006186760, 30772507128, 86191006746, 241815195292

Offset: 0

Colin Defant, Apr 28 2019

nonn

Essentially the same as A217333 and A025273. - R. J. Mathar, May 17 2019

Colin Defant, Motzkin intervals and valid hook configurations, arXiv preprint arXiv:1904.10451 [math.CO], 2019.

Cf. A307786, A307787.

my(x='x+O('x^35)); Vec((1 - 2*x + 2*x^2 - sqrt(1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4))/(2*x^2)) \\ Michel Marcus, May 08 2019

G.f.: (1 - 2*x + 2*x^2 - sqrt(1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4)) / (2*x^2).

D-finite with recurrence: (n+2)*a(n) +2*(-2*n-1)*a(n-1) +4*(n-1)*a(n-2) +2*(-2*n+5)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, May 17 2019

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A217333 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-x)^k ).

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A307788 Number of valid hook configurations of permutations of [n] that avoid the patterns 231 and 321.

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