A242351 -id:A242351 - cp's OEIS frontend

A242352 Number T(n,k) of isoscent sequences of length n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=n+2-ceiling(2*sqrt(n+1)), read by rows.

1, 1, 2, 4, 1, 9, 6, 21, 29, 2, 51, 124, 28, 127, 499, 241, 10, 323, 1933, 1667, 216, 1, 835, 7307, 10142, 2765, 98, 2188, 27166, 56748, 27214, 2637, 22, 5798, 99841, 299485, 227847, 44051, 1546, 2, 15511, 363980, 1514445, 1708700, 563444, 46947, 570

Offset: 0

Joerg Arndt and Alois P. Heinz, May 11 2014

An isoscent sequence of length n is an integer sequence [s(1),...,s(n)] with s(1) = 0 and 0 <= s(i) <= 1 plus the number of level steps in [s(1),...,s(i)].
Columns k=0-10 give: A001006, A243474, A243475, A243476, A243477, A243478, A243479, A243480, A243481, A243482, A243483.
Row sums give A000110.
Last elements of rows give A243484.

			T(4,0) = 9: [0,0,0,0], [0,0,0,1], [0,0,0,2], [0,0,0,3], [0,0,1,1], [0,0,1,2], [0,0,2,2], [0,1,1,1], [0,1,1,2].
T(4,1) = 6: [0,0,1,0], [0,0,2,0], [0,0,2,1], [0,1,0,0], [0,1,0,1], [0,1,1,0].
T(5,2) = 2: [0,0,2,1,0], [0,1,0,1,0].
Triangle T(n,k) begins:
:    1;
:    1;
:    2;
:    4,     1;
:    9,     6;
:   21,    29,     2;
:   51,   124,    28;
:  127,   499,   241,    10;
:  323,  1933,  1667,   216,    1;
:  835,  7307, 10142,  2765,   98;
: 2188, 27166, 56748, 27214, 2637, 22;

Joerg Arndt and Alois P. Heinz, Rows n = 0..114, flattened

Cf. A048993 (for counting level steps), A242351 (for counting ascents), A137251 (ascent sequences counting ascents), A238858 (ascent sequences counting descents), A242153 (ascent sequences counting level steps), A083479.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 0$2)):
seq(T(n), n=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[jJean-François Alcover, Feb 09 2015, after Maple *)

A243228 Number of isoscent sequences of length n with exactly two ascents.

Original entry on oeis.org

3, 25, 128, 525, 1901, 6371, 20291, 62407, 187272, 552104, 1606762, 4631643, 13256644, 37742047, 107025452, 302585780, 853556449, 2403702976, 6760469822, 18995826302, 53336990264, 149680752886, 419883986837, 1177504825907, 3301408010791, 9254726751126

Offset: 4

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=2 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 2):
seq(a(n), n=4..35);

b[n_, i_, t_] := b[n, i, t] = If[n < 1, 1, Expand[Sum[ If[j > i, x, 1] *b[n - 1, j, t + If[j == i, 1, 0]], {j, 0, t + 1}]]]; a[n_] := Coefficient [b[n - 1, 0, 0], x, 2]; Table[a[n], {n, 4, 35}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

Recurrence: (3*n^3 - 43*n^2 + 120*n + 20)*a(n) = (21*n^3 - 289*n^2 + 712*n + 400)*a(n-1) - (51*n^3 - 665*n^2 + 1374*n + 1540)*a(n-2) + 4*(12*n^3 - 145*n^2 + 230*n + 435)*a(n-3) - (9*n^3 - 87*n^2 + 26*n + 280)*a(n-4) - 2*(3*n^3 - 34*n^2 + 43*n + 100)*a(n-5). - Vaclav Kotesovec, Aug 27 2014
a(n) ~ c * d^n, where d = 2.8019377358048382524722... is the root of the equation 1 + 3*d - 4*d^2 + d^3 = 0, c = 0.9786935821895919379992... is the root of the equation 1 - 49*c^2 + 49*c^3 = 0. - Vaclav Kotesovec, Aug 27 2014
G.f.: x^4*(3 - 5*x + x^2)*(1 - x - x^2) / ((1 - x)^3*(1 - 2*x)^2*(1 - 4*x + 3*x^2 + x^3)) (conjectured). - Colin Barker, May 05 2019

A243229 Number of isoscent sequences of length n with exactly three ascents.

Original entry on oeis.org

17, 229, 1819, 11172, 58847, 280158, 1242859, 5238042, 21245548, 83685745, 322225735, 1218705577, 4544214608, 16751906196, 61188410692, 221832968059, 799344529621, 2865983103387, 10233713828145, 36419029944617, 129245774064864, 457623216922119

Offset: 6

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 6..100

Column k=3 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 3):
seq(a(n), n=6..35);

b[n_, i_, t_] := b[n, i, t] = If[n < 1, 1, Expand[Sum[
    If[j>i, x, 1] *b[n-1, j, t + If[j == i, 1, 0]], {j, 0, t+1}]]];
a[n_] := Coefficient[b[n - 1, 0, 0], x, 3];
Table[a[n], {n, 6, 35}] (* Jean-François Alcover, Aug 27 2021, after Maple code *)

G.f.: x^6*(17 - 213*x + 1118*x^2 - 3135*x^3 + 4851*x^4 - 3492*x^5 - 262*x^6 + 1707*x^7 + 82*x^8 - 1050*x^9 + 189*x^10 + 297*x^11 - 122*x^12 + 11*x^13 + 3*x^14) / ((1 - x)^4*(1 - 2*x)^3*(1 - 4*x + 3*x^2 + x^3)^2*(1 - 8*x + 21*x^2 - 18*x^3 - 3*x^4 + 5*x^5 + 3*x^6)) (conjectured). - Colin Barker, May 04 2019

A243230 Number of isoscent sequences of length n with exactly four ascents.

Original entry on oeis.org

2, 172, 3048, 33065, 275641, 1945529, 12246616, 70948683, 386155045, 2002805428, 10000176270, 48434098918, 228855387073, 1059660102823, 4824874714701, 21663637862920, 96134931734425, 422409581103433, 1840542075175555, 7962723214375505, 34240149114971395

Offset: 7

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 7..100

Column k=4 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 4):
seq(a(n), n=7..35);

b[n_, i_, t_] := b[n, i, t] = If[n < 1, 1, Expand[Sum[
   If[j > i, x, 1] *b[n - 1, j, t + If[j == i, 1, 0]], {j, 0, t + 1}]]];
a[n_] := Coefficient[b[n - 1, 0, 0], x, 4];
Table[a[n], {n, 7, 35}] (* Jean-François Alcover, Aug 27 2021, after Maple code *)

G.f.: x^7*(2 + 58*x - 3648*x^2 + 72673*x^3 - 842380*x^4 + 6646402*x^5 - 38326607*x^6 + 167767231*x^7 - 568689800*x^8 + 1503499267*x^9 - 3082877524*x^10 + 4785455208*x^11 - 5266898088*x^12 + 3276181912*x^13 + 633509453*x^14 - 3771906622*x^15 + 3792580950*x^16 - 1396200816*x^17 - 686750857*x^18 + 1195721330*x^19 - 1109678200*x^20 + 1163509247*x^21 - 851661184*x^22 + 47666957*x^23 + 460007148*x^24 - 312042368*x^25 - 2183667*x^26 + 84057866*x^27 - 24458487*x^28 - 9560501*x^29 + 5700026*x^30 + 307683*x^31 - 694453*x^32 + 45182*x^33 + 51672*x^34 - 3534*x^35 - 1395*x^36 - 135*x^37) / ((1 - x)^5*(1 - 2*x)^4*(1 - 4*x + 3*x^2 + x^3)^3*(1 - 8*x + 21*x^2 - 18*x^3 - 3*x^4 + 5*x^5 + 3*x^6)^2*(1 - 16*x + 105*x^2 - 361*x^3 + 676*x^4 - 606*x^5 + 68*x^6 + 192*x^7 + 83*x^8 - 157*x^9 - 44*x^10 + 45*x^11 + 15*x^12)) (conjectured). - Colin Barker, May 05 2019

A243231 Number of isoscent sequences of length n with exactly five ascents.

Original entry on oeis.org

53, 2751, 56905, 771451, 8134377, 72508373, 573143602, 4140884203, 27910175578, 178061795837, 1086790574778, 6397136152382, 36542124316824, 203563522508400, 1110216333288652, 5946996009211010, 31369110229193995, 163289639236069520, 840329677373681576

Offset: 9

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 9..100

Column k=5 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 5):
seq(a(n), n=9..35);

b[n_, i_, t_] := b[n, i, t] = If[n < 1, 1, Expand[Sum[
    If[j > i, x, 1] *b[n - 1, j, t + If[j == i, 1, 0]], {j, 0, t + 1}]]];
a[n_] := Coefficient[b[n - 1, 0, 0], x, 5];
Table[a[n], {n, 9, 35}] (* Jean-François Alcover, Aug 27 2021, after Maple code *)

A243232 Number of isoscent sequences of length n with exactly six ascents.

Original entry on oeis.org

7, 1422, 61966, 1419901, 22720053, 288205741, 3105573804, 29654436325, 258037473373, 2086374808221, 15899245803550, 115413232143719, 804608004076949, 5421939004211551, 35497436998569721, 226735499246904329, 1417770877081098164, 8703370306993200829

Offset: 10

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 10..100

Column k=6 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 6):
seq(a(n), n=10..35);

A243233 Number of isoscent sequences of length n with exactly seven ascents.

Original entry on oeis.org

436, 45146, 1830826, 45369299, 826196076, 12168531813, 153430719591, 1718168937922, 17525279293159, 165802561604726, 1474718822704304, 12459599973722036, 100805357136035992, 786035745070774366, 5938016635175121413, 43645066989439738813, 313228673357655892678

Offset: 12

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 12..100

Column k=7 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 7):
seq(a(n), n=12..35);

A243234 Number of isoscent sequences of length n with exactly eight ascents.

Original entry on oeis.org

72, 22702, 1726823, 67907889, 1795511707, 36321261053, 605440292330, 8715982350668, 111865311047525, 1309224069548063, 14207107757575607, 144766250416253170, 1398869193142980670, 12919066626921248461, 114754676548076467734, 985458803547385212585

Offset: 13

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 13..100

Column k=8 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 8):
seq(a(n), n=13..35);

A243235 Number of isoscent sequences of length n with exactly nine ascents.

Original entry on oeis.org

5, 7894, 1224994, 78795771, 3067107659, 85773445272, 1895106633310, 35082703708444, 565885383903338, 8174846220800924, 107914203931103343, 1321652161255079196, 15194953405886326264, 165524034548480907270, 1721272958146506221402, 17191653626679380069342

Offset: 14

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 14..100

Column k=9 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 9):
seq(a(n), n=14..35);

A243236 Number of isoscent sequences of length n with exactly ten ascents.

Original entry on oeis.org

1854, 663390, 72572427, 4229760773, 164869021176, 4846616066912, 115525788526251, 2341346575945183, 41696673590069251, 668272802442309946, 9812868808904564866, 133849038379000364897, 1714455118957116764366, 20802230173239523620513, 240794713489411990019752

Offset: 16

Joerg Arndt and Alois P. Heinz, Jun 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 16..100

Column k=10 of A242351.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 10):
seq(a(n), n=16..35);

cp's OEIS Frontend

A242352 Number T(n,k) of isoscent sequences of length n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=n+2-ceiling(2*sqrt(n+1)), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A243228 Number of isoscent sequences of length n with exactly two ascents.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A243229 Number of isoscent sequences of length n with exactly three ascents.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A243230 Number of isoscent sequences of length n with exactly four ascents.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A243231 Number of isoscent sequences of length n with exactly five ascents.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A243232 Number of isoscent sequences of length n with exactly six ascents.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A243233 Number of isoscent sequences of length n with exactly seven ascents.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A243234 Number of isoscent sequences of length n with exactly eight ascents.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A243235 Number of isoscent sequences of length n with exactly nine ascents.

Views

Author

Keywords

Links