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%I #14 Jul 27 2022 09:51:50
%S 0,1,8,147,5824,405845,43733976,6726601063,1398047697152,
%T 377278848390249,128228860181918440,53585748788874537851,
%U 27001973543813627400768,16144773936121968789213757,11300021011239061076228900024,9152162639827097780662174019535
%N a(n) = Sum_{0<=m<n} Product_{-m<=j<=m} (n-j).
%C n-m | a(n)-a(m) for all n,m.
%H Robert Israel, <a href="/A279127/b279127.txt">Table of n, a(n) for n = 0..225</a>
%H Math StackExchange, <a href="http://math.stackexchange.com/questions/2046098/are-certain-integer-functions-well-defined-modulo-different-primes-necessarily-p/2046134#2046134">Are certain integer functions well-defined modulo different primes necessarily polynomials?</a>
%F a(n) = A003470(2n-1) for n >= 1.
%F a(n) = n*hypergeom([1,n+1,1-n],[],-1).
%F a(n+3) = -a(n)+(4*n^2+6*n-1)*a(n+1)+(4*n^2+18*n+17)*a(n+2)+8*n+12.
%F D-finite with recurrence +(-2*n+5)*a(n) +(2*n-5)*(4*n^2-6*n+1)*a(n-1) -(2*n-1)*(4*n^2-18*n+19)*a(n-2) +(2*n-1)*a(n-3)=0. - _R. J. Mathar_, Jul 27 2022
%p f:= n -> add(mul(n-m,m=-k..k),k=0..n):
%p map(f, [$0..40]);
%t Table[Sum[Product[n - j, {j, -m, m}], {m,0,n}], {n,0,25}] (* _G. C. Greubel_, Dec 07 2016 *)
%o (PARI) a(n) = sum(m=0, n-1, prod(j=-m, m, n-j)); \\ _Michel Marcus_, Dec 07 2016
%Y Cf. A003470.
%K nonn
%O 0,3
%A _Robert Israel_, Dec 06 2016
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