A094729
Number of connected ordered 2-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 7, 37, 151, 541, 1807, 5797, 18151, 55981, 171007, 519157, 1569751, 4733821, 14250607, 42850117, 128746951, 386634061, 1160688607, 3483638677, 10454061751, 31368476701, 94118013007, 282379204837, 847187946151, 2541664501741, 7625194831807
Offset: 0
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With[{nmax = 50}, CoefficientList[Series[Exp[3*x] - 3*Exp[2*x] + 4*Exp[x] - 2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{6,-11,6},{0,1,1,7},30] (* Harvey P. Dale, Aug 07 2023 *)
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x='x+O('x^50); concat([0], Vec(serlaplace(exp(3*x)-3*exp(2*x) +4*exp(x)-2))) \\ G. C. Greubel, Oct 06 2017
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concat(0, Vec(x*(1 - 5*x + 12*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Oct 13 2017
A094730
Number of connected ordered 3-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 25, 337, 4321, 46681, 437305, 3721537, 29740561, 228000361, 1699890985, 12435686737, 89792976001, 642488104441, 4567920215065, 32331017955937, 228106608326641, 1605738151030921, 11285298643841545, 79223419486529137
Offset: 0
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With[{nmax = 50}, CoefficientList[Series[Exp[7*x] - 6*Exp[5*x] + 3*Exp[4*x] + 17*Exp[3*x] - 30*Exp[2*x] + 21*Exp[x] - 6, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 08 2017 *)
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x='x+O('x^50); concat([0], Vec(serlaplace(exp(7*x) - 6*exp(5*x) + 3*exp(4*x) + 17*exp(3*x) - 30*exp(2*x) + 21*exp(x) - 6))) \\ G. C. Greubel, Oct 08 2017
A094731
Number of connected ordered 4-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 79, 2101, 63991, 1841461, 45677479, 986583781, 19210969591, 347527345621, 5968468471879, 98788140462661, 1592387628858391, 25181074712937781, 392680081411090279, 6061279724768728741, 92859536016650958391, 1414764491802643937941
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..845
- Index entries for linear recurrences with constant coefficients, signature (63,-1701,25887,-245427,1510257,-6084119,15754053,-24891552,21416940,-7484400).
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With[{nmax = 50}, CoefficientList[Series[Exp[15*x] - 12*Exp[11*x] + 24*Exp[9*x] - 8*Exp[7*x] + 27*Exp[6*x] - 96*Exp[5*x] - 6*Exp[4*x] + 246*Exp[3*x] - 288*Exp[2*x] + 138*Exp[x] - 26, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 08 2017 *)
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x='x+O('x^50); concat([0], Vec(serlaplace(exp(15*x) -12*exp(11*x) +24*exp(9*x) -8*exp(7*x) +27*exp(6*x) -96*exp(5*x) -6*exp(4*x) +246*exp(3*x) -288*exp(2*x) +138*exp(x) -26))) \\ G. C. Greubel, Oct 08 2017
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concat(0, Vec(x*(1 - 62*x + 1717*x^2 - 27062*x^3 + 285547*x^4 - 1926074*x^5 + 8088135*x^6 - 28645362*x^7 + 105534360*x^8 - 194594400*x^9) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 9*x)*(1 - 11*x)*(1 - 15*x)) + O(x^30))) \\ Colin Barker, Oct 13 2017
A094732
Number of connected ordered 5-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 241, 11761, 736801, 50524321, 3176975761, 171220124881, 8021076673921, 337296669440641, 13098877345981681, 479949442942292401, 16851170646696553441, 573314381587074123361, 19054886956855687698001
Offset: 0
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With[{nmax = 50}, CoefficientList[Series[Exp[31*x] - 20*Exp[23*x] + 60*Exp[19*x] + 20*Exp[17*x] + 5*Exp[16*x] - 95*Exp[15*x] - 120*Exp[14*x] + 150*Exp[13*x] + 180*Exp[12*x] - 420*Exp[11*x] - 110*Exp[10*x] + 620*Exp[9*x] + 160*Exp[8*x] - 690*Exp[7*x] + 840*Exp[6*x] - 936*Exp[5*x] - 1140*Exp[4*x] + 3560*Exp[3*x] - 3010*Exp[2*x] + 1095*Exp[x] - 150, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 08 2017 *)
A094733
Number of connected ordered 6-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 727, 64357, 7512151, 1143589261, 177189092767, 23695071256837, 2668384623898951, 260281239918269821, 22750998388694399407, 1832528834698360763317, 138901315742774351716951, 10061570091146133148587181, 704453005976484684303395647
Offset: 0
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With[{nmax = 50}, CoefficientList[Series[Exp[63*x] - 30*Exp[47*x] + 120*Exp[39*x] + 60*Exp[35*x] + 60*Exp[33*x] - 18*Exp[32*x] - 324*Exp[31*x] - 720*Exp[29*x] + 810*Exp[27*x] + 120*Exp[26*x] + 480*Exp[25*x] + 480*Exp[24*x] - 900*Exp[23*x] - 720*Exp[22*x] - 240*Exp[21*x] - 900*Exp[20*x] + 2640*Exp[19*x] + 615*Exp[18*x] + 480*Exp[17*x] + 510*Exp[16*x] - 2955*Exp[15*x] - 5070*Exp[14*x] + 3960*Exp[13*x] + 7320*Exp[12*x] - 8640*Exp[11*x] - 4860*Exp[10*x] + 10620*Exp[9*x] + 9210*Exp[8*x] - 21910*Exp[7*x] + 16290*Exp[6*x] - 120*Exp[5*x] - 34470*Exp[4*x] + 53925*Exp[3*x] - 34950*Exp[2*x] + 10208*Exp[x] - 1082, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 08 2017 *)
A094734
Number of connected 2-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 4, 19, 76, 271, 904, 2899, 9076, 27991, 85504, 259579, 784876, 2366911, 7125304, 21425059, 64373476, 193317031, 580344304, 1741819339, 5227030876, 15684238351, 47059006504, 141189602419, 423593973076, 1270832250871, 3812597415904, 11437993574299
Offset: 0
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Join[{0},LinearRecurrence[{6,-11,6},{1,1,4},30]] (* Harvey P. Dale, Nov 28 2014 *)
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x='x+O('x^50); concat([0], Vec(serlaplace((1/2!)*(exp(3*x) - 3*exp(2*x) + 5*exp(x) - 3)))) \\ G. C. Greubel, Oct 07 2017
A094735
Number of connected 3-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 8, 75, 796, 8051, 73788, 623155, 4965836, 38028051, 283400668, 2072874035, 14966280876, 107083717651, 761327161148, 5388524417715, 38017832427916, 267623218488851, 1880883687651228, 13203904989574195, 92616374066478956, 649261556308773651
Offset: 0
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Table[(7^n-6*5^n+20*3^n+3*4^n-39*2^n+35)/6(1-UnitStep[-n]), {n, 0, 20}] (* Benedict W. J. Irwin, May 25 2016 *)
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x='x+O('x^50); concat([0], Vec(serlaplace((1/3!)*(exp(7*x) - 6*exp(5*x) + 3*exp(4*x) + 20*exp(3*x) - 39*exp(2*x) + 35*exp(x) - 14)))) \\ G. C. Greubel, Oct 08 2017
A094736
Number of connected 4-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 13, 189, 3816, 88646, 2013383, 42040699, 807900526, 14537331816, 249111237453, 4119281678909, 66371933499236, 1049372070568186, 16362812045380723, 252561404639492319, 3869204360738213946, 58948921926491795756, 894453362388005059193
Offset: 0
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Table[(314 - 501*2^n + 3*2^(2 + 2 n) + 359*3^n + 2^n*3^(3 + n) + 8*3^(1 + 2 n) - 132*5^n - 2*7^n - 12*11^n + 15^n)/ 24 (1 - UnitStep[-n]), {n, 0, 20}] (* Benedict W. J. Irwin, May 25 2016 *)
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concat(0, Vec(x*(1-62*x +1651*x^2 -24816*x^3 +233562*x^4 -1431634*x^5 +5791471*x^6 -15717948*x^7 +28663875*x^8 -28066500*x^9) / ((1 -x)*(1 -2*x)*(1 -3*x)*(1 -4*x)*(1 -5*x)*(1 -6*x)*(1 -7*x)*(1 -9*x)*(1 -11*x)*(1 -15*x)) + O(x^50))) \\ Colin Barker, May 25 2016
A094737
Number of connected 5-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 19, 387, 12796, 588332, 30409555, 1510137553, 68451901642, 2839832714238, 109655179461961, 4007814663515939, 140559147215148208, 4779718456846032064, 158823449312897655487, 5186933187595033751445
Offset: 0
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Table[ (3434 - 1095*4^n + 5*16^n - 3545*2^(n + 1) + 5*2^(3 n + 5) + 6665*3^n + 860*9^n + 5*4^(n + 1)*3^(n + 2) - 2106*5^n - 17*3^n*5^(n + 1) + 185*6^(n + 1) - 15*2^(n + 3)*7^n - 15*7^(n + 2) - 11*10^(n + 1) - 540*11^n + 150*13^n + 20*17^n + 60*19^n - 20*23^n + 31^n)/120 (1 - UnitStep[-n]), {n, 0, 20}] (* Benedict W. J. Irwin, May 25 2016 *)
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