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Revision History for A002003

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A002003

a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k).
(history; published version)

#103 by Harvey P. Dale at Wed Sep 18 12:50:36 EDT 2024

STATUS

editing

approved

#102 by Harvey P. Dale at Wed Sep 18 12:50:32 EDT 2024

MATHEMATICA

Table[2*Sum[Binomial[n-1, k]Binomial[n+k, k], {k, 0, n-1}], {n, 0, 30}] (* Harvey P. Dale, Sep 18 2024 *)

STATUS

approved

editing

#101 by N. J. A. Sloane at Wed Mar 22 21:59:19 EDT 2023

STATUS

proposed

approved

#100 by Jon E. Schoenfield at Wed Mar 22 21:57:23 EDT 2023

STATUS

editing

proposed

#99 by Jon E. Schoenfield at Wed Mar 22 21:57:21 EDT 2023

FORMULA

a(n) = 2 * JacobiP(n-1,0,1,3) = ((7*n+3)*LegendreP(n,3) - (n+1)*LegendreP(n+1,3)) /(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010

a(n) = Hyper2F1([-n, n], [1], -1) for n > 0. - Peter Luschny, Aug 02 2014

a(n) = A001850(n) - A002002(n), for n > 0. - Shel Kaphan, Feb 15 2023

STATUS

proposed

editing

#98 by Chai Wah Wu at Wed Mar 22 20:56:00 EDT 2023

STATUS

editing

proposed

#97 by Chai Wah Wu at Wed Mar 22 20:55:55 EDT 2023

PROG

def A002003(n): return sum(comb(n, k)**2*k<<k -1 for k in range(1, n+1))//n <<1 if n else 0 # Chai Wah Wu, Mar 22 2023

STATUS

proposed

editing

#96 by Chai Wah Wu at Wed Mar 22 20:45:01 EDT 2023

STATUS

editing

proposed

#95 by Chai Wah Wu at Wed Mar 22 20:44:51 EDT 2023

PROG

def A002003(n): return sum(comb(n, k)**2*k<<k for k in range(1, n+1))//n if n else 0 # Chai Wah Wu, Mar 22 2023

STATUS

proposed

editing

#94 by Chai Wah Wu at Wed Mar 22 20:41:58 EDT 2023

STATUS

editing

proposed