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A112951
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a(n) = number of indecomposable Schur rings over the group Z_{2^n}.
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1
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1, 2, 5, 16, 63, 271, 1225, 5726, 27461, 134461, 669795, 3384945, 17316771, 89518347, 466932059, 2454546192, 12990743783, 69164599115, 370186756425, 1990638982239, 10749412063853, 58265968105385, 316903203993921
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OFFSET
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1,2
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COMMENTS
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Counts also special lattice paths combining ones enumerated by the Catalan numbers A000108 and the large Schroeder numbers A006318.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Changed field to group in the name of the sequence.
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STATUS
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approved
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