A036239 -id:A036239 - cp's OEIS frontend

A051303 Number of 3-element proper antichains of an n-element set.

0, 0, 0, 1, 30, 605, 9030, 110901, 1200150, 11932285, 111885510, 1006471301, 8786447670, 75039565965, 630534185190, 5234341175701, 43059373189590, 351805681631645, 2859550165976070, 23152657123816101, 186907026783617910, 1505512392025329325

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (29,-343,2135,-7504,14756,-14832,5760).

Cf. A032263, A036239, A051112.

[(8^n - 9*6^n + 15*5^n - 4*4^n - 9*3^n + 8*2^n - 2) / Factorial(3) : n in [0..25]]; // G. C. Greubel, Oct 06 2017

A051303:=n->(8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!: seq(A051303(n), n=0..30); # Wesley Ivan Hurt, Oct 06 2017

Table[(8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!, {n,0,25}] (* G. C. Greubel, Oct 06 2017 *)

for(n=0,25, print1((8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2 )/3!, ", ")) \\ G. C. Greubel, Oct 06 2017

a(n) = (8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!.
G.f.: x^3*(360*x^3-78*x^2-x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). - Colin Barker, Nov 27 2012
a(n) = 29*a(n-1) - 343*a(n-2) + 2135*a(n-3) - 7504*a(n-4) + 14756*a(n-5) - 14832*a(n-6) + 5760*a(n-7) for n > 6. - Wesley Ivan Hurt, Oct 06 2017
E.g.f.: exp(x)*(exp(x) - 1)^3*(2 - 2*exp(x) - 3*exp(2*x) + 3*exp(3*x) + exp(4*x))/6. - Stefano Spezia, Sep 28 2024

A051304 Number of 4-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 5, 780, 41545, 1442910, 39400305, 923889960, 19550316665, 384954289170, 7196416532305, 129495073447740, 2264887575116985, 38775513868485030, 653195404307491505, 10869004241198535120, 179171681947204584505, 2932562923651659410490, 47737465871974206925905

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A032263, A036239, A051112.

[(16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/(24): n in [0..50]]; // G. C. Greubel, Oct 07 2017

Table[(16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/4!, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,50, print1((16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/4!, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = (1/4!) * (16^n -18*12^n +60*10^n -9*9^n -102*8^n +105*7^n -90*6^n +95*5^n -31*4^n -33*3^n +28*2^n -6).

G.f. x^4*( 5 +365*x -7935*x^2 +46885*x^3 -191420*x^4 +2285460*x^5 -14380560*x^6 +27216000*x^7 ) / ( (x-1) *(9*x-1) *(6*x-1) *(7*x-1) *(3*x-1) *(5*x-1) *(2*x-1) *(12*x-1) *(10*x-1) *(4*x-1) *(8*x-1) *(16*x-1) ). - R. J. Mathar, Jun 13 2013

Terms a(16) onward added by G. C. Greubel, Oct 07 2017

A051305 Number of 5-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 0, 543, 118629, 12564636, 907001550, 51751693161, 2527016053023, 110737868741742, 4489929936371880, 171944175793168779, 6309813148166785257, 224210698542088771968

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

G. C. Greubel, Table of n, a(n) for n = 0..660

Cf. A032263, A036239, A051112.

[(32^n - 30*24^n + 150*20^n - 45*18^n + 85*17^n - 515*16^n -450*15^n + 1365*14^n + 390*13^n - 1680*12^n - 22*11^n + 1875*10^n - 1080*9^n - 685*8^n + 980*7^n - 669*6^n + 575*5^n - 195*4^n - 150*3^n + 124*2^n - 24)/(120): n in [0..50]]; // G. C. Greubel, Oct 07 2017

Table[(32^n - 30*24^n + 150*20^n - 45*18^n + 85*17^n - 515*16^n -450*15^n + 1365*14^n + 390*13^n - 1680*12^n - 22*11^n + 1875*10^n - 1080*9^n - 685*8^n + 980*7^n - 669*6^n + 575*5^n - 195*4^n - 150*3^n + 124*2^n - 24)/5!, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,50, print1((32^n - 30*24^n + 150*20^n - 45*18^n + 85*17^n - 515*16^n -450*15^n + 1365*14^n + 390*13^n - 1680*12^n - 22*11^n + 1875*10^n - 1080*9^n - 685*8^n + 980*7^n - 669*6^n + 575*5^n - 195*4^n - 150*3^n + 124*2^n - 24)/5!, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = (1/5!)*(32^n -30*24^n +150*20^n -45*18^n +85*17^n -515*16^n -450*15^n +1365*14^n +390*13^n -1680*12^n -22*11^n +1875*10^n -1080*9^n -685*8^n +980*7^n -669*6^n +575*5^n -195*4^n -150*3^n +124*2^n -24).

A051306 Number of 6-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 0, 300, 233821, 78501094, 15532759830, 2213672795040, 254206334062527, 25146386270836578, 2235664320306737320, 183782806231396191820, 14248056393984957136593

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

G. C. Greubel, Table of n, a(n) for n = 0..550

Cf. A032263, A036239, A051112.

Table[(64^n - 45*48^n + 300*40^n - 135*36^n + 510*34^n - 198*33^n - 1499*32^n - 2700*30^n + 6615*28^n + 1215*27^n - 780*26^n + 3750*25^n - 6750*24^n - 8280*23^n + 3828*22^n - 12285*21^n + 19425*20^n + 31635*19^n - 30105*18^n - 34425*17^n + 24770*16^n + 13125*15^n - 3885*14^n + 390*13^n - 5670*12^n - 12485*11^n + 28575*10^n - 16560*9^n - 3435*8^n + 7868*7^n - 4995*6^n + 3800*5^n - 1301*4^n - 822*3^n + 668*2^n - 120)/6!, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

a(n) = (1/6!)*(64^n -45*48^n +300*40^n -135*36^n +510*34^n -198*33^n -1499*32^n -2700*30^n +6615*28^n +1215*27^n -780*26^n +3750*25^n -6750*24^n -8280*23^n +3828*22^n -12285*21^n +19425*20^n +31635*19^n -30105*18^n -34425*17^n +24770*16^n +13125*15^n -3885*14^n +390*13^n -5670*12^n -12485*11^n +28575*10^n -16560*9^n -3435*8^n +7868*7^n -4995*6^n +3800*5^n -1301*4^n -822*3^n +668*2^n -120).

A379706 Number of nonempty labeled antichains of subsets of [n] such that the largest subset is of size 2.

Original entry on oeis.org

0, 0, 1, 10, 97, 1418, 40005, 2350474, 286192257, 71213783154, 35883905262757, 36419649682704418, 74221659280476132145, 303193505953871645554778, 2480118046704094643352342117, 40601989176407026666590990389338, 1329877330167226219547875498464450945, 87134888326188320631048795061602782878050

Offset: 0

John Tyler Rascoe, Dec 30 2024

			a(2) = 1: {{1,2}}.
a(3) = 10: {{1,2}}, {{1,3}}, {{2,3}}, {{1,2},{3}}, {{1,3},{2}}, {{2,3},{1}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.

Cf. A000225, A000372, A036239, A126883, A379707, (column k=2 of A379712).

from math import comb
def A379706(n): return sum(comb(n,i)*(2**comb(n-i,2) - 1) for i in range(n-1))

a(n) = Sum_{i=0..n-2} binomial(n,i) * (2^binomial(n-i,2) - 1).

A005530 Number of Boolean functions of n variables from Post class F(8,inf); number of degenerate Boolean functions of n variables.

Original entry on oeis.org

2, 6, 38, 942, 325262, 25768825638, 129127208425774833206, 2722258935367507707190488025630791841374

Offset: 1

N. J. A. Sloane, R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 129.

Alois P. Heinz, Table of n, a(n) for n = 1..12
T. E. Allen, J. Goldsmith, N. Mattei, Counting, Ranking, and Randomly Generating CP-nets, 2014.
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974
Y. Raekow and K. Ziegler, A taxonomy of non-cooperatively computable functions, Presented at WEWoRC 2011 (link to conference record).
I. Tomescu, Excerpts from "Introducese in Combinatorica" (1972), pp. 230-1, 44-5, 128-9. (Annotated scanned copy)
Index entries for sequences related to Boolean functions

a(n) = 2^(2^n) - A000371(n). Cf. A036239, A036240.

Sum[(-1)^(j + 1) Binomial[n, j] 2^2^(n - j), {j, 1, n}]

for(n=1,10, print1(sum(j=1,n, (-1)^(j+1)*binomial(n,j)*2^(2^(n-j))), ", ")) \\ G. C. Greubel, Oct 06 2017

a(n) = Sum_{j=1..n} (-1)^(j+1)*binomial(n,j)*2^(2^(n-j)).

More terms from Vladeta Jovovic, Goran Kilibarda

A051307 Number of 7-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 0, 135, 329205, 365924948, 205640068950, 75013516525425, 20611869786684495, 4661763066154503606, 917701003163074793520, 163180081989646991509955, 26889766005753182579964345, 4182467653250525215771670424, 622388054953695081193665509610

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

Cf. A032263, A036239, A051112.

a(n) = 1/7! * (128^n -63*96^n +525*80^n -315*72^n +1785*68^n -1386*66^n +455*65^n -3486*64^n -9450*60^n +21315*56^n +8505*54^n -13650*52^n -5355*51^n +36750*50^n -5145*49^n -14805*48^n -57960*46^n -4725*45^n +45738*44^n +36120*43^n -191835*42^n +43050*41^n +74725*40^n -73710*39^n +333165*38^n +104895*37^n -73395*36^n -54390*35^n -354144*34^n -423192*33^n +383621*32^n +143220*31^n -292425*30^n +753855*29^n +181545*28^n -314685*27^n -114660*26^n -916125*25^n -268716*24^n +1998493*23^n +140833*22^n -2359350*21^n +458675*20^n +2147950*19^n -961758*18^n -1428000*17^n +933380*16^n +578175*15^n -614362*14^n +143052*13^n +45990*12^n -244860*11^n +356475*10^n -199521*9^n -12244*8^n +64778*7^n -40026*6^n +28035*5^n -9604*4^n -5292*3^n +4248*2^n -720).

A051363 Number of 6-element families of an n-element set such that every 3 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 0, 112, 40286, 5485032, 534844548, 45066853496, 3538771308282, 267882021563464, 19861835713621616, 1453175611052688600, 105278656040052332838, 7564280930105061931496, 539399446172552069053404

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

G. C. Greubel, Table of n, a(n) for n = 0..550

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

a(n) = (1/6!)*(64^n -20*56^n +90*52^n +30*50^n +25*49^n -420*48^n -180*47^n +450*46^n +60*45^n +615*44^n +1683*43^n -3252*42^n -6030*41^n +8520*40^n +10560*39^n -15849*38^n -13005*37^n +26410*36^n +10655*35^n -50385*34^n +33390*33^n +29480*32^n -82010*31^n +91215*30^n -67380*29^n +36870*28^n -15249*27^n +4380*26^n -1215*25^n +1390*24^n -695*23^n -1574*22^n +3255*21^n -3075*20^n +1800*19^n -675*18^n +150*17^n +70*16^n -340*14^n +510*13^n -340*12^n +85*11^n -225*8^n +225*7^n +274*4^n -274*3^n -120*2^n +120).

A051364 Number of 5-element families of an n-element set such that every 3 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 0, 225, 21571, 1174122, 51441824, 2038356243, 76714338477, 2804947403364, 100732231517698, 3572491367063421, 125474030774355263, 4371052010746528926, 151172238539268318372

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

G. C. Greubel, Table of n, a(n) for n = 0..660

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

Table[1/5! (32^n - 10*28^n + 30*26^n + 5*25^n - 80*24^n + 45*23^n + 105*22^n - 217*21^n + 205*20^n - 120*19^n + 45*18^n - 10*17^n - 9*16^n + 40*14^n - 60*13^n + 40*12^n - 10*11^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = (1/5!)*(32^n - 10*28^n + 30*26^n + 5*25^n - 80*24^n + 45*23^n + 105*22^n - 217*21^n + 205*20^n - 120*19^n + 45*18^n - 10*17^n - 9*16^n + 40*14^n - 60*13^n + 40*12^n - 10*11^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24).

A051365 Number of 4-element families of an n-element set such that every 3 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 3, 275, 8475, 192385, 3831093, 71466675, 1285857975, 22632300245, 392522268633, 6734698919575, 114576024346875, 1935649374363705, 32505459713369373, 543014736097852475, 9029329231317194175, 149522990698790644765, 2466942184607949641313

Offset: 0

Vladeta Jovovic, Goran Kilibarda

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

[(16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[1/4! (16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6)/24, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/4!)*(16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6).

G.f.: -x^3*(47062848*x^7 -42816008*x^6 +13976678*x^5 -2170583*x^4 +168932*x^3 -5672*x^2 +2*x +3) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(7*x -1)*(8*x -1)*(11*x -1)*(12*x -1)*(13*x -1)*(14*x -1)*(16*x -1)). - Colin Barker, Jul 12 2013

More terms from Colin Barker, Jul 12 2013

cp's OEIS Frontend

A051303 Number of 3-element proper antichains of an n-element set.

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A051304 Number of 4-element proper antichains of an n-element set.

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A051305 Number of 5-element proper antichains of an n-element set.

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A051306 Number of 6-element proper antichains of an n-element set.

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A379706 Number of nonempty labeled antichains of subsets of [n] such that the largest subset is of size 2.

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A005530 Number of Boolean functions of n variables from Post class F(8,inf); number of degenerate Boolean functions of n variables.

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A051307 Number of 7-element proper antichains of an n-element set.

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A051363 Number of 6-element families of an n-element set such that every 3 members of the family have a nonempty intersection.

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