A238858 -id:A238858 - cp's OEIS frontend

A241877 Number of ascent sequences of length n with exactly seven descents.

262, 45355, 2945144, 115640368, 3310089672, 76098190354, 1488843551061, 25742974021760, 403792980134432, 5855533942758206, 79621493875268300, 1026297720267259815, 12647422247984279889, 150029397902618780880, 1722644140748876548506, 19232187878731653646068

Offset: 12

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 12..1000

Column k=7 of A238858.

gf:= -(17168622184562688000*x^30 -165990345532165324800*x^29 +764265306648034344960*x^28 -2227477724647806173184*x^27 +4607560717763655094272*x^26 -7188029219365307265024*x^25 +8771965813694591513856*x^24 -8569977967582670585472*x^23
+6800372189878230880624*x^22 -4416996852693953991344*x^21 +2351482055430886572808*x^20 -1019029358088416724440*x^19 +351892979022335268927*x^18 -91463393523130541293*x^17 +14605225422499318735*x^16 +581363497155871899*x^15 -1354208397644673655*x^14 +543643959424172265*x^13 -143184185751363124*x^12 +28224011083685669*x^11
-4287423192624419*x^10 +497037468594090*x^9 -41750353408807*x^8 +2131740277605*x^7 -1413256993*x^6 -10610071423*x^5 +985051074*x^4 -46399717*x^3 +1072084*x^2 -2911*x -262) *x^12 / ((9*x-1) *(8*x-1)^2 *(7*x-1)^3 *(6*x-1)^4 *(5*x-1)^5 *(4*x-1)^6 *(x-1)^6 *(3*x-1)^7 *(2*x-1)^8):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=12..30);

G.f.: see Maple program.

Recurrence: a(n) + 1834933472251084800000*a(n-42) - 29890483744407552000000*a(n-41) + 234674273308869918720000*a(n-40) - 1183460488011095998464000*a(n-39) + 4310616672232759649894400*a(n-38) - 12090110137133949555572736*a(n-37) + 27184553419658691145826304*a(n-36) - 50373565574321929969631232*a(n-35) + 78480773039085711009583104*a(n-34) - 104366600043561766882639872*a(n-33) + 119864288120509558700623872*a(n-32) - 120001551815106232058621952*a(n-31) + 105512719130330131388301056*a(n-30) - 81976491762435391083895296*a(n-29) + 56559443885408831033142528*a(n-28) - 34795705814009881313016960*a(n-27) + 19151295749903563509912816*a(n-26) - 9455656062744697462579008*a(n-25) + 4196934270184509793587272*a(n-24) - 1677368435971517310106656*a(n-23) + 604352766672278616765729*a(n-22) - 196445323903680067600818*a(n-21) + 57626984175715546403919*a(n-20) - 15254915763530534131260*a(n-19) + 3642342102661923055475*a(n-18) - 783681290841137732154*a(n-17) + 151735847111641722501*a(n-16) - 26388336799044975888*a(n-15) + 4112001903716894106*a(n-14) - 572369284218905028*a(n-13) + 70895600662640518*a(n-12) - 7777305431924328*a(n-11) + 751228057187814*a(n-10) - 63431687262564*a(n-9) + 4639853686722*a(n-8) - 290658484080*a(n-7) + 15363592789*a(n-6) - 671812362*a(n-5) + 23646363*a(n-4) - 643644*a(n-3) + 12711*a(n-2) - 162*a(n-1). - Fung Lam, May 06 2014

A241878 Number of ascent sequences of length n with exactly eight descents.

Original entry on oeis.org

76, 35209, 4039194, 242899690, 9885744698, 308499975033, 7934181861432, 176105387674796, 3481327075812644, 62693622772259358, 1045913922174474652, 16373263083718686437, 242940967901333077989, 3444033483761421576832, 46951392190123945806932

Offset: 13

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Recurrence is of 52 order with constant coefficients (see link below).

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 13..1000
Fung Lam, Recurrence formula for A241878

Column k=8 of A238858.

gf:= -(52553412112007194214400000*x^39 -613289908725672396718080000*x^38 +3446340062218318299267072000*x^37 -12409743778368324350902272000*x^36 +32144544101639652703103877120*x^35 -63739623224842771994751467520*x^34 +100526156498120293361669455872*x^33 -129326381850884353971886854144*x^32 +138078181730156438217858923520*x^31 -123788362154790905517444399360*x^30
+93876678485180434126091194176*x^29 -60428403800498502691099934400*x^28 +32981578627042728537763538064*x^27 -15149960622696298673159858800*x^26 +5745076159441010911548301660*x^25 -1713942189473347432159782004*x^24 +344138671691755284367114047*x^23 -5839405428890160900553740*x^22
-32824622557763063660057790*x^21 +18176377898703093820133728*x^20 -6522560269430279741005099*x^19 +1814942458271772835455036*x^18 -411537276001965041120674*x^17 +77191887852380143515467*x^16 -11948748170525701008430*x^15 +1497118354975682972561*x^14
-144192168126331827895*x^13 +9057386760969303803*x^12 -33939433225045648*x^11 -78967551758852587*x^10 +11947165597637555*x^9 -1090933373158527*x^8 +69435729107323*x^7 -3048021102033*x^6 +80222524613*x^5 -392852647*x^4 -54444097*x^3 +1857643*x^2 -18717*x -76) *x^13 / ((10*x-1) *(9*x-1)^2 *(8*x-1)^3 *(7*x-1)^4 *(6*x-1)^5 *(5*x-1)^6 *(4*x-1)^7 *(x-1)^7 *(3*x-1)^8 *(2*x-1)^9):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=13..30);

G.f.: see Maple program.

A241879 Number of ascent sequences of length n with exactly nine descents.

Original entry on oeis.org

13, 21366, 4578065, 429702983, 25034354941, 1061920763452, 35851255343461, 1018448322393201, 25273317588814845, 562618587861007357, 11458003977913501433, 216682231415882284146, 3849500164272613169698, 64843210660491264824949, 1043390054793718867256585

Offset: 14

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Recurrence is of 63 order with constant coefficients (see link below). - Fung Lam, May 07 2014

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 14..950
Joerg Arndt and Alois P. Heinz, G.f. and Maple program for A241879
Fung Lam, Recurrence formula for A241879

Column k=9 of A238858.

A241880 Number of ascent sequences of length n with exactly ten descents.

Original entry on oeis.org

1, 10030, 4319226, 648334495, 54586013987, 3157647587689, 139956094477219, 5080107725520158, 157820345727409092, 4328163957695091489, 107185085092451745955, 2438347032119886770511, 51637001422611301449267, 1028741824925149047195935, 19445597194074945790332367

Offset: 15

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Recurrence is of 75 order with constant coefficients (see link below). - Fung Lam, May 07 2014

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 15..900
Joerg Arndt and Alois P. Heinz, G.f. and Maple program for A241880
Fung Lam, Recurrence formula for A241880

Column k=10 of A238858.

A241881 Number of ascent sequences of length n with the maximal number of descents.

Original entry on oeis.org

1, 1, 2, 1, 7, 4, 1, 48, 26, 8, 1, 594, 262, 76, 13, 1, 10030, 3571, 933, 169, 19, 1, 205271, 61206, 14351, 2550, 323, 26, 1, 4910802, 1263620, 267378, 45321, 5918, 559, 34, 1, 134636523, 30534920, 5873492, 939681, 121689, 12257, 901, 43, 1, 4166817191

Offset: 0

Joerg Arndt and Alois P. Heinz, May 01 2014

nonn

a(n*(n+1)/2) = a(A000217(n)) = 1.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..140

b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(
      `if`(ji, 1, 0)), j=0..t+1)))
    end:
a:= n-> (p-> coeff(p, x, degree(p)))(b(n, -1$2)):
seq(a(n), n=0..40);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[ji, 1, 0]], {j, 0, t+1}]]]; a[n_] := Function[{p}, Coefficient[p, x, Exponent[ p, x ]]][b[n, -1, -1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 13 2015, after Maple *)

a(n) = A238858(n,Re(n-floor((sqrt(8*n-7)+1)/2))).

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A241877 Number of ascent sequences of length n with exactly seven descents.

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A241878 Number of ascent sequences of length n with exactly eight descents.

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A241879 Number of ascent sequences of length n with exactly nine descents.

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A241880 Number of ascent sequences of length n with exactly ten descents.

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A241881 Number of ascent sequences of length n with the maximal number of descents.

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