A242352 -id:A242352 - cp's OEIS frontend

A242351 Number T(n,k) of isoscent sequences of length n with exactly k ascents; triangle T(n,k), n>=0, 0<=k<=n+3-ceiling(2*sqrt(n+2)), read by rows.

1, 1, 1, 1, 1, 4, 1, 11, 3, 1, 26, 25, 1, 57, 128, 17, 1, 120, 525, 229, 2, 1, 247, 1901, 1819, 172, 1, 502, 6371, 11172, 3048, 53, 1, 1013, 20291, 58847, 33065, 2751, 7, 1, 2036, 62407, 280158, 275641, 56905, 1422, 1, 4083, 187272, 1242859, 1945529, 771451, 61966, 436

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: A000012, A000295, A243228, A243229, A243230, A243231, A243232, A243233, A243234, A243235, A243236.
Row sums give A000110.
Last elements of rows give A243237.

			T(4,0) = 1: [0,0,0,0].
T(4,1) = 11: [0,0,0,1], [0,0,0,2], [0,0,0,3], [0,0,1,0], [0,0,1,1], [0,0,2,0], [0,0,2,1], [0,0,2,2], [0,1,0,0], [0,1,1,0], [0,1,1,1].
T(4,2) = 3: [0,0,1,2], [0,1,0,1], [0,1,1,2].
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   11,     3;
  1,   26,    25;
  1,   57,   128,    17;
  1,  120,   525,   229,     2;
  1,  247,  1901,  1819,   172;
  1,  502,  6371, 11172,  3048,   53;
  1, 1013, 20291, 58847, 33065, 2751, 7;
  ...

Joerg Arndt and Alois P. Heinz, Rows n = 0..100, flattened

Cf. A048993 (for counting level steps), A242352 (for counting descents), 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>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
T:= n-> (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[j>i, x, 1]*b[n-1, j, t + If[j == i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 09 2015, after Maple *)

A243474 Number of isoscent sequences of length n with exactly one descent.

Original entry on oeis.org

1, 6, 29, 124, 499, 1933, 7307, 27166, 99841, 363980, 1319404, 4763927, 17155264, 61672791, 221499015, 795198010, 2854898575, 10253237150, 36846414395, 132518215788, 477049025009, 1719101735260, 6201858101192, 22399768386170, 80998670324341, 293244129636085

Offset: 3

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

			a(4) = 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].

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=1 of A242352.

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[jJean-François Alcover, Feb 09 2015, after A242352 *)

Recurrence: 2*(n+1)*(n+2)*(2*n+3)*(12397*n^7 - 189057*n^6 + 1186699*n^5 - 4027875*n^4 + 7966576*n^3 - 8920548*n^2 + 4726368*n - 164160)*a(n) = 2*(n+1)*(2*n + 1)*(74382*n^8 - 997975*n^7 + 5169531*n^6 - 12939205*n^5 + 13804539*n^4 + 4032932*n^3 - 23655372*n^2 + 20014848*n - 1477440)*a(n-1) - 2*(347116*n^9 - 3797013*n^8 + 13426236*n^7 - 10697862*n^6 - 45689304*n^5 + 115458855*n^4 - 47561392*n^3 - 85364460*n^2 + 57311424*n - 1477440)*a(n-2) - (1822359*n^10 - 28485611*n^9 + 180879786*n^8 - 605859318*n^7 + 1141835871*n^6 - 1127112699*n^5 + 285267312*n^4 + 592120604*n^3 - 783881808*n^2 + 409889664*n - 50388480)*a(n-3) - 2*(247940*n^10 - 5628293*n^9 + 49022694*n^8 - 212356554*n^7 + 479773884*n^6 - 459074385*n^5 - 250049558*n^4 + 1020252416*n^3 - 830684880*n^2 + 423719136*n - 432293760)*a(n-4) + 2*(n-4)*(2070299*n^9 - 30481583*n^8 + 181205557*n^7 - 572698754*n^6 + 1060137133*n^5 - 1157719883*n^4 + 582047111*n^3 + 378941580*n^2 - 897279300*n + 403878960)*a(n-5) + 6*(n-5)*(n-4)*(768614*n^8 - 9316516*n^7 + 43281239*n^6 - 101853145*n^5 + 131895047*n^4 - 75435871*n^3 - 41445228*n^2 + 118112292*n - 101468592)*a(n-6) + 117*(n-6)*(n-5)*(n-4)*(12397*n^7 - 102278*n^6 + 312694*n^5 - 496340*n^4 + 374821*n^3 + 103402*n^2 - 440568*n + 590400)*a(n-7). - Vaclav Kotesovec, Jun 06 2014

a(n) ~ c * d^n / n^(3/2), where d = 1/6*(847+33*sqrt(33))^(1/3) + 44/(3*(847+33*sqrt(33))^(1/3)) + 2/3 = 3.802619145513318... is the root of the equation 4*d^3 - 8*d^2 - 24*d - 13 = 0 and c = sqrt(2565 + 2*(3*(692007507 - 5151139*sqrt(33)))^(1/3) + 2*(3*(692007507 + 5151139*sqrt(33)))^(1/3)) / (4*sqrt(21*Pi)) = 2.695007157151120689163873119078514352395445402... . - Vaclav Kotesovec, Jun 06 2014, updated Mar 16 2024

A243475 Number of isoscent sequences of length n with exactly two descents.

Original entry on oeis.org

2, 28, 241, 1667, 10142, 56748, 299485, 1514445, 7415873, 35424718, 165963977, 765639729, 3488740303, 15739196772, 70434524941, 313136541466, 1384706596078, 6096641331963, 26747810596747, 117015840972949, 510745056446585, 2225207050547044, 9680854941464722

Offset: 5

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

			a(5) = 2: [0,0,2,1,0], [0,1,0,1,0].

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 5..100

Column k=2 of A242352.

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[jJean-François Alcover, Feb 09 2015, after A242352 *)

A243476 Number of isoscent sequences of length n with exactly three descents.

Original entry on oeis.org

10, 216, 2765, 27214, 227847, 1708700, 11832896, 77170252, 480381209, 2881934792, 16782041642, 95373448420, 531211249132, 2909490548577, 15712454516343, 83849134367589, 442957532385072, 2319975180476948, 12061609189508662, 62313958408668146, 320192732763382270

Offset: 7

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

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

Column k=3 of A242352.

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, 7, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)

A243477 Number of isoscent sequences of length n with exactly four descents.

Original entry on oeis.org

1, 98, 2637, 44051, 563444, 6054955, 57523592, 498540949, 4024955854, 30719216646, 224034574434, 1573966189886, 10719427410360, 71119561115093, 461496506508538, 2938387657219301, 18406158143420637, 113681266064450777, 693570688210367081, 4186481577282095644

Offset: 8

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 8..100

Column k=4 of A242352.

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, 8, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)

A243478 Number of isoscent sequences of length n with exactly five descents.

Original entry on oeis.org

22, 1546, 46947, 944701, 14745521, 193273378, 2228848403, 23305050081, 225560761204, 2051114040040, 17719591217257, 146674616051322, 1171101886353096, 9067551569835697, 68378670584901547, 504001041740002317, 3641720452674969516, 25859746275607624792

Offset: 10

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

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

Column k=5 of A242352.

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, 10, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)

A243479 Number of isoscent sequences of length n with exactly six descents.

Original entry on oeis.org

2, 570, 34643, 1139465, 26298831, 479669603, 7386945631, 100099764774, 1227140160022, 13879804132399, 146957303797639, 1472647088474040, 14087369324642004, 129522280702233381, 1150891407181302417, 9927963973481921295, 83454334937159452579

Offset: 11

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 11..100

Column k=6 of A242352.

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, 6]; Table[a[n], {n, 11, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)

A243480 Number of isoscent sequences of length n with exactly seven descents.

Original entry on oeis.org

130, 18196, 1027497, 35855578, 920132845, 19024724767, 334824869566, 5199938158961, 73073359931844, 946309985456857, 11449696018832655, 130815643776548834, 1423250326223813866, 14845562100052915605, 149281547622659687547, 1453772869428542126084

Offset: 13

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

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

Column k=7 of A242352.

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, 7]; Table[a[n], {n, 13, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)

A243481 Number of isoscent sequences of length n with exactly eight descents.

Original entry on oeis.org

17, 6867, 710088, 38504985, 1409180462, 39390143719, 903369758170, 17794252503829, 310620860769359, 4914610198933595, 71670394607559276, 975829774811610622, 12530984703191609205, 153000363388748917171, 1787972768823076763190, 20107368357857569858580

Offset: 14

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

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

Column k=8 of A242352.

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, 8]; Table[a[n], {n, 14, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)

A243482 Number of isoscent sequences of length n with exactly nine descents.

Original entry on oeis.org

1, 1842, 381540, 33260687, 1764436385, 67244544426, 2018462900473, 50530007996596, 1096301608172492, 21183116277395043, 371926458625987333, 6025302522233251390, 91147815762874899354, 1299862836761209689164, 17611186926254063273931, 228129250314546958924423

Offset: 15

Joerg Arndt and Alois P. Heinz, Jun 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 15..100

Column k=9 of A242352.

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, 9]; Table[a[n], {n, 15, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)

cp's OEIS Frontend

A242351 Number T(n,k) of isoscent sequences of length n with exactly k ascents; triangle T(n,k), n>=0, 0<=k<=n+3-ceiling(2*sqrt(n+2)), read by rows.

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A243474 Number of isoscent sequences of length n with exactly one descent.

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A243475 Number of isoscent sequences of length n with exactly two descents.

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A243476 Number of isoscent sequences of length n with exactly three descents.

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A243477 Number of isoscent sequences of length n with exactly four descents.

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A243478 Number of isoscent sequences of length n with exactly five descents.

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A243479 Number of isoscent sequences of length n with exactly six descents.

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A243480 Number of isoscent sequences of length n with exactly seven descents.

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A243481 Number of isoscent sequences of length n with exactly eight descents.

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A243482 Number of isoscent sequences of length n with exactly nine descents.