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A007901 Number of minimal unavoidable n-celled pebbling configurations.

%I A007901 #32 May 19 2024 14:15:19
%S A007901 0,0,0,0,4,22,98,412,1700,6974,28576,117146,480722,1974914,8122084,
%T A007901 33435390,137757480,567998152,2343472004,9674252070,39956606552,
%U A007901 165099840920,682446679582,2821858504062,11671572244666,48287711006032,199822535773958,827069530795224,3423890026639184,14176516741276534
%N A007901 Number of minimal unavoidable n-celled pebbling configurations.
%D A007901 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.50.
%H A007901 Alois P. Heinz, <a href="/A007901/b007901.txt">Table of n, a(n) for n = 1..1000</a>
%H A007901 F. R. K. Chung, R. L. Graham, J. A. Morrison and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/pebbling.pdf">Pebbling a chessboard</a>, Amer. Math. Monthly 102 (1995), pp. 113-123.
%H A007901 Google Labs, <a href="http://www.7427466391.com/">Google Labs congratulations puzzle</a>
%H A007901 Marcus Kazmierczak, <a href="http://www.mkaz.com/math/google/">Google Labs Puzzles</a>, Jul 29, 2004.
%H A007901 Slashdot (CmdrTaco), <a href="http://developers.slashdot.org/article.pl?sid=04/09/16/139232&amp;tid=217&amp;tid=8">Google's Math Puzzle</a>, Thu Sep 16, 2004.
%F A007901 G.f.: x^3*((1-3*x+x^2)*sqrt(1-4*x)-1+5*x-x^2-6*x^3)/(1-7*x+14*x^2-9*x^3) [from the Stanley reference]. - _Joerg Arndt_, Apr 20 2011
%F A007901 Conjecture: (n-3)*(n-8)*a(n) +(-11*n^2+127*n-324)*a(n-1) +42*(n^2-12*n+34)*a(n-2) +(-65*n^2+799*n-2400)*a(n-3) +18*(n-6)*(2*n-13)*a(n-4)=0. - _R. J. Mathar_, Aug 14 2012
%p A007901 The Maple snippet provides an alternative solution to the Google congratulations puzzle at http://www.7427466391.com. After running the Maple code, f(1) to f(4) match the puzzle, with f(5) being 1510865746 and f(6) being 6171783928.
%p A007901 Digits:=2000: E:=evalf(exp(1)): g:=n->trunc((E-(10^(-n)*trunc(E*10^n)))*10^(10+n)): h:=[0,0,0,0,4,22,98,412,1700]: f:=k->g(h[k+3]):
%t A007901 p[k_] := If[k < 7, {0, 4, -14, 22, -20, 6}[[k]], 0]; h[n_] := Sum[ k*p[k] * Binomial[2*n-k-1, n-k], {k, 1, n}]/n; u[n_] := Sum[ Sum[Binomial[j, n-k-j]*7^(2*k-n+j)*Binomial[k, j]*(-2)^(-n+k+2*j)*3^(2*(n-k-j)), {j, 0, k}], {k, 0, n}]; b[n_] := Sum[h[i]*u[n-i], {i, 1, n}]; a[n_] := If[n<2, 0 , b[n-2]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jan 14 2015, after _Vladimir Kruchinin_ *)
%t A007901 CoefficientList[Series[x^3((1-3x+x^2)Sqrt[1-4x]-1+5x-x^2-6x^3)/(1-7x+14x^2-9x^3),{x,0,30}],x] (* _Harvey P. Dale_, May 19 2024 *)
%o A007901 (PARI) x='x+O('x^44);
%o A007901 gf=x^3*((1-3*x+x^2)*sqrt(1-4*x)-1+5*x-x^2-6*x^3)/(1-7*x+14*x^2-9*x^3);
%o A007901 Vec(gf) \\ _Joerg Arndt_, Apr 20 2011
%o A007901 (Maxima)
%o A007901 Polynom:[0,4,-14,22,-20,6];
%o A007901 p(k):=if k<7 then Polynom[k] else 0;
%o A007901 h(n):=sum(k*p(k)*binomial(2*n-k-1,n-k),k,1,n)/n;
%o A007901 u(n):=sum(sum(binomial(j,n-k-j)*7^(2*k-n+j)*binomial(k,j)*(-2)^(-n+k+2*j)*3^(2*(n-k-j)),j,0,k),k,0,n);
%o A007901 b(n):=sum(h(i)*u(n-i),i,1,n);
%o A007901 a(n):=if n<2 then 0 else b(n-2);
%o A007901 makelist(a(n),n,0,40); /* _Vladimir Kruchinin_, Sep 20 2014 */
%Y A007901 Cf. A007902.
%K A007901 nonn,easy,nice
%O A007901 1,5
%A A007901 odlyzko(AT)dtc.umn.edu (A. M. Odlyzko)