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A112951 a(n) = number of indecomposable Schur rings over the group Z_{2^n}. 1
1, 2, 5, 16, 63, 271, 1225, 5726, 27461, 134461, 669795, 3384945, 17316771, 89518347, 466932059, 2454546192, 12990743783, 69164599115, 370186756425, 1990638982239, 10749412063853, 58265968105385, 316903203993921 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Counts also special lattice paths combining ones enumerated by the Catalan numbers A000108 and the large Schroeder numbers A006318.
LINKS
I. Kovács, The number of indecomposable Schur rings over a cyclic 2-group, Séminaire Lotharingien de Combinatoire, vol. 51 (2005), Article B51h.
FORMULA
G.f.: x * (2/(3*x + sqrt(1 - 4*x) + sqrt(1 - 6*x + x^2)) + x / (1-x)).
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
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
MATHEMATICA
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 *)
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];
Table[(-1)^n Binomial[-1, -2 + n] UnitStep[-2 + n] + 2 G[n], {n, 1, 20}] (* Benedict W. J. Irwin, May 29 2016 *)
PROG
(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
(Maxima)
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;
makelist(a(n), n, 0, 17); /* Vladimir Kruchinin, Mar 14 2016 */
CROSSREFS
Sequence in context: A159603 A058117 A007124 * A124470 A105072 A022494
KEYWORD
nonn
AUTHOR
Valery A. Liskovets, Oct 10 2005
EXTENSIONS
Changed field to group in the name of the sequence.
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)