A112951 - OEIS

%I #43 May 25 2017 04:18:23

%S 1,2,5,16,63,271,1225,5726,27461,134461,669795,3384945,17316771,

%T 89518347,466932059,2454546192,12990743783,69164599115,370186756425,

%U 1990638982239,10749412063853,58265968105385,316903203993921

%N a(n) = number of indecomposable Schur rings over the group Z_{2^n}.

%C Counts also special lattice paths combining ones enumerated by the Catalan numbers A000108 and the large Schroeder numbers A006318.

%H G. C. Greubel, <a href="/A112951/b112951.txt">Table of n, a(n) for n = 1..1000</a>

%H I. Kovács, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s51kovacs.html">The number of indecomposable Schur rings over a cyclic 2-group</a>, Séminaire Lotharingien de Combinatoire, vol. 51 (2005), Article B51h.

%F G.f.: x * (2/(3*x + sqrt(1 - 4*x) + sqrt(1 - 6*x + x^2)) + x / (1-x)).

%F a(n) ~ 2*sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^(n-1) / ((9-6*sqrt(2) + sqrt(8*sqrt(2)-11))^2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jan 26 2015

%F a(n) = (-1)^n Binomial(-1,n-2) + 2*y(n), n>1, where y(n) is the recurrence function as defined in the Mathematica code below. - _Benedict W. J. Irwin_, May 29 2016

%t CoefficientList[Series[2/(3*x + Sqrt[1-4*x] + Sqrt[1-6*x+x^2]) + x/(1-x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jan 26 2015 *)

%t G[k_] := DifferenceRoot[Function[{y, n}, {-3584 n (1 + n) (1 + 2 n) (3 + 2 n) y[n] + (2486400 + 6395008 n + 5886976 n^2 + 2296832 n^3 + 318464 n^4) y[1 + n] + (-149581920 - 226382448 n - 125114096 n^2 - 29606592 n^3 - 2471104 n^4) y[2 + n] + (2392388400 + 2496057788 n + 937406996 n^2 + 146051392 n^3 + 7435024 n^4) y[3 + n] + (-15092019120 - 11076799292 n - 2735859684 n^2 - 231411748 n^3 - 980556 n^4) y[4 + n] + (28910784480 + 10141803126 n - 744863243 n^2 - 592663986 n^3 - 53858857 n^4) y[5 + n] + (89507964600 + 84246427620 n + 27245631360 n^2 + 3720321660 n^3 + 184192200 n^4) y[6 + n] + (-356568849840 - 235250921386 n - 57446553587 n^2 - 6169678274 n^3 - 246355873 n^4) y[7 + n] + (481079425200 + 268744660228 n + 56030192026 n^2 + 5170045052 n^3 + 178223894 n^4) y[8 + n] + (-348798457920 - 172159785960 n - 31778032340 n^2 - 2600336400 n^3 - 79601140 n^4) y[9 + n] + (154841013840 + 68888499880 n + 11467407120 n^2 + 846581720 n^3 + 23388480 n^4) y[10 + n] + (-44493790080 - 18055052404 n - 2741443748 n^2 - 184609376 n^3 - 4652152 n^4) y[11 + n] + (8452013280 + 3151713492 n + 439754444 n^2 + 27211428 n^3 + 630076 n^4) y[12 + n] + (-1057845360 - 364182910 n - 46917105 n^2 - 2680670 n^3 - 57315 n^4) y[13 + n] + 4 (13 + n) (14 + n) (116265 + 19819 n + 844 n^2) y[14 + n] - (13 + n) (14 + n) (15 + n) (1464 + 119 n) y[15 + n] + 2 (13 + n) (14 + n) (15 + n) (16 + n) y[16 + n] == 0, y[0] == 0, y[1] == 1/2, y[2] == 1/2, y[3] == 2, y[4] == 15/2, y[5] == 31, y[6] == 135, y[7] == 612, y[8] == 5725/2, y[9] == 13730, y[10] == 67230, y[11] == 334897, y[12] == 1692472, y[13] == 8658385, y[14] == 44759173, y[15] == 233466029}]][k];

%t Table[(-1)^n Binomial[-1, -2 + n] UnitStep[-2 + n] + 2 G[n], {n, 1, 20}] (* _Benedict W. J. Irwin_, May 29 2016 *)

%o (PARI) default(seriesprecision, 30); Vec(2/(3*x + sqrt(1-4*x) + sqrt(1-6*x+x^2)) + x/(1-x) + O(x^30)) \\ _Michel Marcus_, Jan 26 2015

%o (Maxima)

%o a(n):=(sum((m*(sum(((sum(binomial(j+m,j-i)*binomial(j+i+m-1,j+m-1),i,0,j))* sum((k*binomial(m+k,k)*binomial(2*(n-k-j-m),n-j-m))/(n-k-j-m),k,1,(n-j-m)/2))/ (j+m),j,0,n-m-1))+(m*sum(binomial(n,n-m-i)*binomial(n+i-1,n-1),i,0,n-m))/n) ,m,1,n))+(sum((k*binomial(2*(n-k),n))/(n-k),k,1,n/2))+1;

%o makelist(a(n),n,0,17); /* _Vladimir Kruchinin_, Mar 14 2016 */

%Y Cf. A000108, A006318.

%K nonn

%O 1,2

%A _Valery A. Liskovets_, Oct 10 2005

%E Changed field to group in the name of the sequence.