A243366 -id:A243366 - cp's OEIS frontend

A243752 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.

1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 11, 2, 9, 16, 12, 4, 1, 1, 57, 69, 5, 127, 161, 98, 35, 7, 1, 323, 927, 180, 1515, 1997, 1056, 280, 14, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 10455, 25638, 18357, 4115, 220, 1, 20705, 68850, 77685, 34840, 5685, 246, 1

Offset: 0

Alois P. Heinz, Jun 09 2014

			Triangle T(n,k) begins:
: n\k :    0     1     2     3    4    5  ...
+-----+----------------------------------------------------------
:  0  :    1;                                 [row  0 of A131427]
:  1  :    0,    1;                           [row  1 of A131427]
:  2  :    0,    1,    1;                     [row  2 of A090181]
:  3  :    1,    3,    1;                     [row  3 of A001263]
:  4  :    1,   11,    2;                     [row  4 of A091156]
:  5  :    9,   16,   12,    4,   1;          [row  5 of A091869]
:  6  :    1,   57,   69,    5;               [row  6 of A091156]
:  7  :  127,  161,   98,   35,   7,   1;     [row  7 of A092107]
:  8  :  323,  927,  180;                     [row  8 of A091958]
:  9  : 1515, 1997, 1056,  280,  14;          [row  9 of A135306]
: 10  : 4191, 5539, 3967, 1991, 781, 244, ... [row 10 of A094507]

Alois P. Heinz, Rows n = 0..270, flattened

Columns k=0-10 give: A243754, A243770, A243771, A243772, A243773, A243774, A243775, A243776, A243777, A243778, A243779, or main diagonals of A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
Row sums give A000108.
Cf. A098978, A114463, A114848, A116424, A135305, A242450, A243366, A243838.

A243412 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 112, 352, 1136, 3742, 12529, 42513, 145868, 505234, 1764157, 6203370, 21947490, 78072209, 279062937, 1001803617, 3610366030, 13057141261, 47373444827, 172381857939, 628944880851, 2300410562946, 8433110899963, 30980398420830, 114034887644860

Offset: 0

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=0 of A243366.

Column k=45 of A243753.

Recurrence: (n+1)*(n+2)*(817*n^7 - 24387*n^6 + 285094*n^5 - 1647261*n^4 + 4787137*n^3 - 5628540*n^2 - 1552284*n + 6122952)*a(n) = (n+1)*(1634*n^8 - 47957*n^7 + 542786*n^6 - 2900786*n^5 + 6449435*n^4 + 3292426*n^3 - 41693904*n^2 + 63681552*n - 24491808)*a(n-1) + 3*n*(2451*n^8 - 73161*n^7 + 850153*n^6 - 4796076*n^5 + 12712261*n^4 - 7403931*n^3 - 33886709*n^2 + 64848252*n - 30495792)*a(n-2) - (8170*n^9 - 256125*n^8 + 3222045*n^7 - 20734872*n^6 + 70290303*n^5 - 101053185*n^4 - 62925628*n^3 + 384515340*n^2 - 387509328*n + 86320944)*a(n-3) + 3*(4085*n^9 - 134190*n^8 + 1787518*n^7 - 12351340*n^6 + 46074358*n^5 - 78991732*n^4 - 20763151*n^3 + 311152124*n^2 - 443676900*n + 188645328)*a(n-4) - (8170*n^9 - 280635*n^8 + 3929664*n^7 - 28666521*n^6 + 113672493*n^5 - 215520840*n^4 + 17606573*n^3 + 648300408*n^2 - 951192216*n + 363243312)*a(n-5) + 2*(4085*n^9 - 146445*n^8 + 2159949*n^7 - 16771674*n^6 + 71463813*n^5 - 145058547*n^4 - 9273941*n^3 + 640553178*n^2 - 1114925472*n + 598040712)*a(n-6) + (8170*n^9 - 305145*n^8 + 4669113*n^7 - 37343346*n^6 + 161916525*n^5 - 325736907*n^4 - 55373986*n^3 + 1484026824*n^2 - 2345628420*n + 1080273456)*a(n-7) + (6536*n^9 - 253920*n^8 + 4039503*n^7 - 33528057*n^6 + 150519924*n^5 - 315037869*n^4 - 26105741*n^3 + 1400728128*n^2 - 2351058696*n + 1235710944)*a(n-8) + (n-9)*(6536*n^8 - 204900*n^7 + 2511339*n^6 - 14959584*n^5 + 41778954*n^4 - 25451829*n^3 - 129319352*n^2 + 282520572*n - 168563664)*a(n-9) + 3*(n-10)*(n-9)*(817*n^7 - 18668*n^6 + 155929*n^5 - 559001*n^4 + 589888*n^3 + 1351597*n^2 - 3752130*n + 2343528)*a(n-10). - Vaclav Kotesovec, Jun 05 2014

a(n) ~ c * d^n / n^(3/2), where d = 3.8821590268628506747194368909643384060073824... is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c = 0.56162811676670317653498040062091920282038218... . - Vaclav Kotesovec, Jun 05 2014

A243413 Number of Dyck paths of semilength n having exactly 1 occurrence of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 5, 19, 70, 259, 962, 3585, 13399, 50201, 188481, 709001, 2671624, 10082895, 38107919, 144214978, 546413880, 2072553851, 7869081412, 29904874545, 113744129791, 432969825404, 1649313815911, 6287005845873, 23980562901849, 91523321091182, 349497990760012

Offset: 4

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..550
Vaclav Kotesovec, Recurrence (of order 10)

Column k=1 of A243366.

a(n) ~ c * d^n / sqrt(n), where d = 3.8821590268628506747194368909643384... (same as for A243412) is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c = 0.0159763870992602878106411532836296... . - Vaclav Kotesovec, Jun 05 2014

A243414 Number of Dyck paths of semilength n having exactly 2 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 7, 34, 149, 627, 2584, 10529, 42606, 171563, 688255, 2752912, 10985005, 43747708, 173937910, 690592594, 2738547328, 10848121023, 42931655341, 169759128539, 670744883641, 2648384384709, 10450336782375, 41212385684767, 162440029038575, 639946101535124

Offset: 6

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..550
Vaclav Kotesovec, Recurrence (of order 10)

Column k=2 of A243366.

a(n) ~ c * d^n * sqrt(n), where d = 3.8821590268628506747194368909643384... (same as for A243412) is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c = 0.000227236615409194082906635273578... . - Vaclav Kotesovec, Jun 05 2014

A243415 Number of Dyck paths of semilength n having exactly 3 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 9, 54, 279, 1334, 6092, 27048, 117896, 507173, 2160151, 9128266, 38326830, 160063712, 665442560, 2755685897, 11372798741, 46794914849, 192029181971, 786126469079, 3211253947191, 13091799706905, 53277016211285, 216451122516387, 878040805034630

Offset: 8

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..550

Column k=3 of A243366.

A243416 Number of Dyck paths of semilength n having exactly 4 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 11, 79, 474, 2561, 12937, 62524, 293146, 1344602, 6065734, 27007870, 118984043, 519589872, 2252141576, 9699519424, 41541816291, 177048628294, 751293699026, 3175665502731, 13376228005419, 56162636663430, 235124074987923, 981718571224412, 4088916793922566

Offset: 10

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..550

Column k=4 of A243366.

A243417 Number of Dyck paths of semilength n having exactly 5 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 13, 109, 748, 4531, 25228, 132515, 667747, 3263146, 15576803, 72995158, 336986263, 1536565549, 6933515253, 31007787862, 137599702886, 606467289880, 2656919897648, 11577389890752, 50204031478108, 216750954307831, 932068704988570, 3993428974519241

Offset: 12

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 12..550

Column k=5 of A243366.

A243418 Number of Dyck paths of semilength n having exactly 6 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 15, 144, 1115, 7509, 45864, 261185, 1412742, 7350116, 37106514, 182907585, 884290974, 4207172844, 19747711113, 91627676418, 420914259695, 1916722429108, 8660949017183, 38866735108620, 173341678278027, 768774806179919, 3392247359157876, 14899045118613987

Offset: 14

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 14..550

Column k=6 of A243366.

A243419 Number of Dyck paths of semilength n having exactly 7 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 17, 184, 1589, 11802, 78666, 484163, 2806039, 15527186, 82851064, 429392845, 2173167944, 10783733773, 52629404231, 253233665755, 1203597734972, 5659518993517, 26361150741197, 121755788054172, 558129404726455, 2541096189931118, 11497990460196322

Offset: 16

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 16..550

Column k=7 of A243366.

A243420 Number of Dyck paths of semilength n having exactly 8 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 19, 229, 2184, 17759, 128509, 851831, 5278129, 31020976, 174803309, 952033480, 5042006351, 26087126992, 132340281386, 660143136637, 3245280843311, 15751929981822, 75602686035094, 359256881535197, 1691965976395069, 7904617778358794, 36660673151053000

Offset: 18

Alois P. Heinz, Jun 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 18..550

Column k=8 of A243366.

cp's OEIS Frontend

A243752 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.

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A243412 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243413 Number of Dyck paths of semilength n having exactly 1 occurrence of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243414 Number of Dyck paths of semilength n having exactly 2 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243415 Number of Dyck paths of semilength n having exactly 3 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243416 Number of Dyck paths of semilength n having exactly 4 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243417 Number of Dyck paths of semilength n having exactly 5 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243418 Number of Dyck paths of semilength n having exactly 6 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243419 Number of Dyck paths of semilength n having exactly 7 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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A243420 Number of Dyck paths of semilength n having exactly 8 (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

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