The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A001871

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A001871

Expansion of 1/(1 - 3*x + x^2)^2.
(history; published version)

#138 by Michael De Vlieger at Wed Nov 06 10:19:22 EST 2024

STATUS

reviewed

approved

#137 by Joerg Arndt at Wed Nov 06 08:57:02 EST 2024

STATUS

proposed

reviewed

#136 by Peter Bala at Wed Nov 06 06:37:44 EST 2024

STATUS

editing

proposed

#135 by Peter Bala at Tue Nov 05 17:19:26 EST 2024

FORMULA

a(n) = -a(-4-n) = ((4n4*n+2)*F(2n2*n) + (7n7*n+5)*F(2n2*n+1))/5 with F(n) = A000045 (Fibonacci numbers).

a(n) = (Sum_{k=0..n} S(k, 3)*S(n-k, 3)) , where S(n, x) = U(n, x/2) is the n-th Chebyshev polynomials polynomial of the 2nd kind, A049310. - Paul Barry, Nov 14 2003

a(n) = Sum_{k=1..n+1} F(2k)*F(2(n-k+2)) , where F(k) is the k-th Fibonacci number. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008

a(n) = Sum_{k = 0..n} (n+2*k+1)*binomial(n+k, 2*k) ? - _From _Peter Bala_, Nov 02 05 2024: (Start)

a(n) = Sum_{k = 0..n} (n + 2*k + 1)*binomial(n+k, 2*k).

a(n) = (n+1) * hypergeom([-n, n+1, (n+3)/2], [1/2, (n+1)/2], -1/4).

Second-order recurrence: n*a(n) = 3*(n + 1)*a(n-1) - (n + 2)*a(n-2) with a(0) = 1 and a(1) = 6. (End)

#134 by Peter Bala at Sat Nov 02 06:48:45 EDT 2024

FORMULA

a(n) = Sum_{k = 0..n} (n+2*k+1)*binomial(n+k, 2*k) ? - Peter Bala, Nov 02 2024

STATUS

approved

editing

#133 by Michael De Vlieger at Tue Aug 06 16:55:46 EDT 2024

STATUS

reviewed

approved

#132 by Stefano Spezia at Tue Aug 06 16:54:05 EDT 2024

STATUS

proposed

reviewed

Discussion

Tue Aug 06

16:54

Stefano Spezia: You are welcome

16:55

Michael De Vlieger: Going to go back and edit the seqs already approved with full stop instead of comma. The brilliance of the editing system! Better OEIS!

#131 by Stefano Spezia at Tue Aug 06 16:53:44 EDT 2024

STATUS

editing

proposed

Discussion

Tue Aug 06

16:54

Michael De Vlieger: Thanks Stefano, will correct others, hold on.

#130 by Stefano Spezia at Tue Aug 06 16:53:42 EDT 2024

LINKS

Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck. , <a href="http://jl.baril.u-bourgogne.fr/bmrs.pdf">Catalan words avoiding a pattern of length four</a>, Univ. de Bourgogne (France, 2024). See p. 5.

STATUS

approved

editing

#129 by Michael De Vlieger at Tue Aug 06 16:53:30 EDT 2024

STATUS

reviewed

approved