cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A302897 Number of permutations of [n] having exactly four alternating descents.

Original entry on oeis.org

16, 117, 1056, 8699, 76840, 704834, 6847224, 70145634, 758805360, 8650769675, 103790273632, 1308042935717, 17285090008280, 239069573622720, 3454877140757160, 52081336911782580, 817705146857293920, 13351745346381390705, 226414777897783513040
Offset: 5

Views

Author

Alois P. Heinz, Apr 15 2018

Keywords

Comments

Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

Examples

			a(5) = 16: 21435, 21534, 31425, 31524, 32415, 32514, 41325, 41523, 42315, 42513, 43512, 51324, 51423, 52314, 52413, 53412.

Links

Crossrefs

Column k=5 of A145876.

Programs

  • Maple
    b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 6)
    end:
a:= n-> coeff(b(n, 0), x, 5):
seq(a(n), n=5..30);
  • Mathematica
    nmax = 30; Drop[CoefficientList[Series[((x^4 - 12*x^3 + 60*x^2 - 168*x + 48*Sin[x] + 96)*Cos[x]^2 - ((x^3 - 4*x^2 + 12*x - 24)*Sin[x] + 5*x^3 - 32*x^2 + 60*x - 24)*x*Cos[x] + (- 6*x^4 + 52*x^3 - 168*x^2 + 192*x - 96)*Sin[x] - 6*x^4 + 44*x^3 - 120*x^2 + 192*x - 96)/(24*Cos[x]^3 + (- 24*Sin[x] + 72)*Cos[x]^2 + (48*Sin[x] - 48)*Cos[x] + 96*Sin[x] - 96), {x, 0, nmax}], x] * Range[0, nmax]!, 5] (* Vaclav Kotesovec, Apr 30 2018 *)

Formula

a(n) ~ (4 - Pi)^4 * 2^(n + 5/2) * n^(n + 9/2) / (4! * Pi^(n + 9/2) * exp(n)). - Vaclav Kotesovec, Apr 29 2018
E.g.f.: ((x^4 - 12*x^3 + 60*x^2 - 168*x + 48*sin(x) + 96)*cos(x)^2 - ((x^3 - 4*x^2 + 12*x - 24)*sin(x) + 5*x^3 - 32*x^2 + 60*x - 24)*x*cos(x) + (- 6*x^4 + 52*x^3 - 168*x^2 + 192*x - 96)*sin(x) - 6*x^4 + 44*x^3 - 120*x^2 + 192*x - 96)/(24*cos(x)^3 + (- 24*sin(x) + 72)*cos(x)^2 + (48*sin(x) - 48)*cos(x) + 96*sin(x) - 96). - Vaclav Kotesovec, Apr 30 2018

A302898 Number of permutations of [n] having exactly five alternating descents.

Original entry on oeis.org

61, 594, 6669, 67054, 704834, 7570716, 84889638, 992272308, 12127342203, 154898419006, 2066994606155, 28788990664242, 418074366639272, 6322807317984024, 99466230062507580, 1625658804523009416, 27571506609797250441, 484700416772477950602, 8822485993502063393465
Offset: 6

Views

Author

Alois P. Heinz, Apr 15 2018

Keywords

Comments

Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

Examples

			a(6) = 61: 214365, 215364, 215463, 216354, ..., 635142, 635241, 645132, 645231.

Links

Crossrefs

Column k=6 of A145876.

Programs

  • Maple
    b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 7)
    end:
a:= n-> coeff(b(n, 0), x, 6):
seq(a(n), n=6..30);
  • Mathematica
    nmax = 30; Drop[CoefficientList[Series[(120*Cos[x]^3 + (x^5 - 10*x^4 + 60*x^3 - 240*x^2 + 600*x - 120*Sin[x] - 360)*Cos[x]^2 + ((- 5*x^4 + 40*x^3 - 180*x^2 + 480*x - 240)*Sin[x] - 25*x^4 + 200*x^3 - 540*x^2 + 480*x - 240)*Cos[x] + (- 13*x^5 + 130*x^4 - 540*x^3 + 1200*x^2 - 1200*x + 480)*Sin[x] - 17*x^5 + 170*x^4 - 660*x^3 + 1200*x^2 - 1200*x + 480)/((120*Sin[x] - 360)*Cos[x]^2 - 480*Sin[x] + 480), {x, 0, nmax}], x] * Range[0, nmax]!, 6] (* Vaclav Kotesovec, Apr 30 2018 *)

Formula

a(n) ~ (4 - Pi)^5 * 2^(n + 5/2) * n^(n + 11/2) / (5! * Pi^(n + 11/2) * exp(n)). - Vaclav Kotesovec, Apr 29 2018
E.g.f.: (120*cos(x)^3 + (x^5 - 10*x^4 + 60*x^3 - 240*x^2 + 600*x - 120*sin(x) - 360)*cos(x)^2 + ((- 5*x^4 + 40*x^3 - 180*x^2 + 480*x - 240)*sin(x) - 25*x^4 + 200*x^3 - 540*x^2 + 480*x - 240)*cos(x) + (- 13*x^5 + 130*x^4 - 540*x^3 + 1200*x^2 - 1200*x + 480)*sin(x) - 17*x^5 + 170*x^4 - 660*x^3 + 1200*x^2 - 1200*x + 480)/((120*sin(x) - 360)*cos(x)^2 - 480*sin(x) + 480). - Vaclav Kotesovec, Apr 30 2018

A302899 Number of permutations of [n] having exactly six alternating descents.

Original entry on oeis.org

272, 3407, 46348, 556952, 6847224, 84889638, 1085246904, 14322115212, 195951082944, 2781436057021, 40985527637668, 626827892111140, 9945795998932920, 163614736611741324, 2788498384849238640, 49195762917367001256, 897689701635240236352, 16927557342294928274187
Offset: 7

Views

Author

Alois P. Heinz, Apr 15 2018

Keywords

Comments

Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

Examples

			a(7) = 272: 2143657, 2143756, 2153647, 2153746, ..., 7561423, 7562314, 7562413, 7563412.

Links

Crossrefs

Column k=7 of A145876.

Programs

  • Maple
    b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 8)
    end:
a:= n-> coeff(b(n, 0), x, 7):
seq(a(n), n=7..30);
  • Mathematica
    nmax = 30; Drop[CoefficientList[Series[(1440*Cos[x]^4 + (- x^6 + 6*x^5 - 30*x^4 + 120*x^3 - 360*x^2 + 720*x)*Cos[x]^3 + ((- x^6 + 18*x^5 - 150*x^4 + 840*x^3 - 3240*x^2 + 7920*x - 4320)*Sin[x] - 29*x^6 + 420*x^5 - 2610*x^4 + 9120*x^3 - 18360*x^2 + 16560*x - 7200)*Cos[x]^2 + 28*x*((x^5 - (129/14)*x^4 + (255/7)*x^3 - 90*x^2 + (900/7)*x - 360/7)*Sin[x] + 31*x^5*(1/14) - 321*x^4*(1/14) + 645*x^3*(1/7) - 1170*x^2*(1/7) + 900*x*(1/7) - 360/7)*Cos[x] + (90*x^6 - 1104*x^5 + 5640*x^4 - 15120*x^3 + 21600*x^2 - 17280*x + 5760)*Sin[x] + 90*x^6 - 1056*x^5 + 5160*x^4 - 13680*x^3 + 21600*x^2 - 17280*x + 5760)/(720*Cos[x]^4 + (720*Sin[x] - 2160)*Cos[x]^3 + (2880*Sin[x] - 5760)*Cos[x]^2 + (- 2880*Sin[x] + 2880)*Cos[x] - 5760*Sin[x] + 5760), {x, 0, nmax}], x] * Range[0, nmax]!, 7] (* Vaclav Kotesovec, Apr 30 2018 *)

Formula

a(n) ~ (4 - Pi)^6 * 2^(n + 5/2) * n^(n + 13/2) / (6! * Pi^(n + 13/2) * exp(n)). - Vaclav Kotesovec, Apr 29 2018
E.g.f.: (1440*cos(x)^4 + (- x^6 + 6*x^5 - 30*x^4 + 120*x^3 - 360*x^2 + 720*x)*cos(x)^3 + ((- x^6 + 18*x^5 - 150*x^4 + 840*x^3 - 3240*x^2 + 7920*x - 4320)*sin(x) - 29*x^6 + 420*x^5 - 2610*x^4 + 9120*x^3 - 18360*x^2 + 16560*x - 7200)*cos(x)^2 + 28*x*((x^5 - (129/14)*x^4 + (255/7)*x^3 - 90*x^2 + (900/7)*x - 360/7)*sin(x) + 31*x^5*(1/14) - 321*x^4*(1/14) + 645*x^3*(1/7) - 1170*x^2*(1/7) + 900*x*(1/7) - 360/7)*cos(x) + (90*x^6 - 1104*x^5 + 5640*x^4 - 15120*x^3 + 21600*x^2 - 17280*x + 5760)*sin(x) + 90*x^6 - 1056*x^5 + 5160*x^4 - 13680*x^3 + 21600*x^2 - 17280*x + 5760)/(720*cos(x)^4 + (720*sin(x) - 2160)*cos(x)^3 + (2880*sin(x) - 5760)*cos(x)^2 + (- 2880*sin(x) + 2880)*cos(x) - 5760*sin(x) + 5760). - Vaclav Kotesovec, Apr 30 2018

A302900 Number of permutations of [n] having exactly seven alternating descents.

Original entry on oeis.org

1385, 21682, 350240, 4945368, 70145634, 992272308, 14322115212, 211595659320, 3216832016019, 50412205403030, 815486339550108, 13622914005990480, 235041722344009380, 4187522527966916520, 77010173788311008040, 1461190162226869057872, 28588437379997078589117
Offset: 8

Views

Author

Alois P. Heinz, Apr 15 2018

Keywords

Comments

Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

Links

Crossrefs

Column k=8 of A145876.

Programs

  • Maple
    b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 9)
    end:
a:= n-> coeff(b(n, 0), x, 8):
seq(a(n), n=8..30);
  • Mathematica
    nmax = 30; Drop[CoefficientList[Series[(5040*Cos[x]^4 + (7*x^6 - 84*x^5 + 630*x^4 - 3360*x^3 + 12600*x^2 - 30240*x + 5040*Sin[x] + 15120)*Cos[x]^3 + ((x^7 - 14*x^6 + 126*x^5 - 840*x^4 + 4200*x^3 - 15120*x^2 + 35280*x - 20160)*Sin[x] + 60*x^7 - 840*x^6 + 5544*x^5 - 23520*x^4 + 67200*x^3 - 120960*x^2 + 105840*x - 40320)*Cos[x]^2 + ((- 196*x^6 + 2352*x^5 - 12600*x^4 + 40320*x^3 - 75600*x^2 + 60480*x - 20160)*Sin[x] - 434*x^6 + 5208*x^5 - 25200*x^4 + 60480*x^3 - 75600*x^2 + 60480*x - 20160)*Cos[x] + (- 298*x^7 + 4172*x^6 - 25200*x^5 + 85680*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)*Sin[x] - 332*x^7 + 4648*x^6 - 27720*x^5 + 90720*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)/(5040*Cos[x]^4 + (20160*Sin[x] - 40320)*Cos[x]^2 - 40320*Sin[x] + 40320), {x, 0, nmax}], x] * Range[0, nmax]!, 8] (* Vaclav Kotesovec, Apr 30 2018 *)

Formula

a(n) ~ (4 - Pi)^7 * 2^(n + 5/2) * n^(n + 15/2) / (7! * Pi^(n + 15/2) * exp(n)). - Vaclav Kotesovec, Apr 29 2018
E.g.f.: (5040*cos(x)^4 + (7*x^6 - 84*x^5 + 630*x^4 - 3360*x^3 + 12600*x^2 - 30240*x + 5040*sin(x) + 15120)*cos(x)^3 + ((x^7 - 14*x^6 + 126*x^5 - 840*x^4 + 4200*x^3 - 15120*x^2 + 35280*x - 20160)*sin(x) + 60*x^7 - 840*x^6 + 5544*x^5 - 23520*x^4 + 67200*x^3 - 120960*x^2 + 105840*x - 40320)*cos(x)^2 + ((- 196*x^6 + 2352*x^5 - 12600*x^4 + 40320*x^3 - 75600*x^2 + 60480*x - 20160)*sin(x) - 434*x^6 + 5208*x^5 - 25200*x^4 + 60480*x^3 - 75600*x^2 + 60480*x - 20160)*cos(x) + (- 298*x^7 + 4172*x^6 - 25200*x^5 + 85680*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)*sin(x) - 332*x^7 + 4648*x^6 - 27720*x^5 + 90720*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)/(5040*cos(x)^4 + (20160*sin(x) - 40320)*cos(x)^2 - 40320*sin(x) + 40320). - Vaclav Kotesovec, Apr 30 2018

A302901 Number of permutations of [n] having exactly eight alternating descents.

Original entry on oeis.org

7936, 151853, 2866632, 46901985, 758805360, 12127342203, 195951082944, 3216832016019, 53984412657360, 928559550102410, 16402837435610856, 297876978668583126, 5564330063809902240, 106938416843133746250, 2114402162990417017920, 43002161983507383542010
Offset: 9

Views

Author

Alois P. Heinz, Apr 15 2018

Keywords

Comments

Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

Links

Crossrefs

Column k=9 of A145876.

Programs

  • Maple
    b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 10)
    end:
a:= n-> coeff(b(n, 0), x, 9):
seq(a(n), n=9..30);
  • Mathematica
    nmax = 30; Drop[CoefficientList[Series[((x^8 - 24*x^7 + 280*x^6 - 2352*x^5 + 15120*x^4 - 73920*x^3 + 262080*x^2 - 604800*x + 80640*Sin[x] + 322560)*Cos[x]^4 - x*((x^7 - 8*x^6 + 56*x^5 - 336*x^4 + 1680*x^3 - 6720*x^2 + 20160*x - 40320)*Sin[x] + 123*x^7 - 1488*x^6 + 8568*x^5 - 34944*x^4 + 105840*x^3 - 228480*x^2 + 302400*x - 120960)*Cos[x]^3 + ((- 124*x^8 + 2456*x^7 - 22064*x^6 + 123984*x^5 - 483840*x^4 + 1310400*x^3 - 2257920*x^2 + 1854720*x - 645120)*Sin[x] - 1136*x^8 + 20088*x^7 - 157472*x^6 + 722736*x^5 - 2116800*x^4 + 3971520*x^3 - 4515840*x^2 + 3144960*x - 967680)*Cos[x]^2 + ((1012*x^8 - 13808*x^7 + 81872*x^6 - 282240*x^5 + 624960*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*Sin[x] + 1508*x^8 - 21472*x^7 + 129808*x^6 - 423360*x^5 + 786240*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*Cos[x] + (2520*x^8 - 40592*x^7 + 286048*x^6 - 1149120*x^5 + 2862720*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)*Sin[x] + 2520*x^8 - 40048*x^7 + 278432*x^6 - 1108800*x^5 + 2782080*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)/(40320*Cos[x]^5 + (- 40320*Sin[x] + 201600)*Cos[x]^4 + (161280*Sin[x] - 322560)*Cos[x]^3 + (483840*Sin[x] - 806400)*Cos[x]^2 + (- 322560*Sin[x] + 322560)*Cos[x] - 645120*Sin[x] + 645120), {x, 0, nmax}], x] * Range[0, nmax]!, 9] (* Vaclav Kotesovec, Apr 30 2018 *)

Formula

a(n) ~ (4 - Pi)^8 * 2^(n + 5/2) * n^(n + 17/2) / (8! * Pi^(n + 17/2) * exp(n)). - Vaclav Kotesovec, Apr 29 2018
E.g.f.: ((x^8 - 24*x^7 + 280*x^6 - 2352*x^5 + 15120*x^4 - 73920*x^3 + 262080*x^2 - 604800*x + 80640*sin(x) + 322560)*cos(x)^4 - x*((x^7 - 8*x^6 + 56*x^5 - 336*x^4 + 1680*x^3 - 6720*x^2 + 20160*x - 40320)*sin(x) + 123*x^7 - 1488*x^6 + 8568*x^5 - 34944*x^4 + 105840*x^3 - 228480*x^2 + 302400*x - 120960)*cos(x)^3 + ((- 124*x^8 + 2456*x^7 - 22064*x^6 + 123984*x^5 - 483840*x^4 + 1310400*x^3 - 2257920*x^2 + 1854720*x - 645120)*sin(x) - 1136*x^8 + 20088*x^7 - 157472*x^6 + 722736*x^5 - 2116800*x^4 + 3971520*x^3 - 4515840*x^2 + 3144960*x - 967680)*cos(x)^2 + ((1012*x^8 - 13808*x^7 + 81872*x^6 - 282240*x^5 + 624960*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*sin(x) + 1508*x^8 - 21472*x^7 + 129808*x^6 - 423360*x^5 + 786240*x^4 - 846720*x^3 + 564480*x^2 - 161280*x)*cos(x) + (2520*x^8 - 40592*x^7 + 286048*x^6 - 1149120*x^5 + 2862720*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)*sin(x) + 2520*x^8 - 40048*x^7 + 278432*x^6 - 1108800*x^5 + 2782080*x^4 - 4515840*x^3 + 4515840*x^2 - 2580480*x + 645120)/(40320*cos(x)^5 + (- 40320*sin(x) + 201600)*cos(x)^4 + (161280*sin(x) - 322560)*cos(x)^3 + (483840*sin(x) - 806400)*cos(x)^2 + (- 322560*sin(x) + 322560)*cos(x) - 645120*sin(x) + 645120). - Vaclav Kotesovec, Apr 30 2018

A302902 Number of permutations of [n] having exactly nine alternating descents.

Original entry on oeis.org

50521, 1160026, 25260211, 473324450, 8650769675, 154898419006, 2781436057021, 50412205403030, 928559550102410, 17440458896525180, 334876925319944690, 6583281405926363500, 132633340608724861210, 2740015852551381054980, 58057801350608276240150
Offset: 10

Views

Author

Alois P. Heinz, Apr 15 2018

Keywords

Comments

Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).
From Vaclav Kotesovec, Apr 29 2018: (Start)
In general, number of permutations of [n] having exactly k alternating descents (column k+1 of A145876) is asymptotic to a(n,k) ~ (4 - Pi)^k * 2^(n + 2) * n^k * n! / (k! * Pi^(n + k + 1)).
Equivalently, a(n,k) ~ (4 - Pi)^k * 2^(n + 5/2) * n^(n + k + 1/2) / (k! * Pi^(n + k + 1/2) * exp(n)).
(End)

Links

Crossrefs

Column k=10 of A145876.

Programs

  • Maple
    b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 11)
    end:
a:= n-> coeff(b(n, 0), x, 10):
seq(a(n), n=10..30);
  • Mathematica
    nmax = 30; Drop[CoefficientList[Series[(362880*Cos[x]^5 + (x^9 - 18*x^8 + 216*x^7 - 2016*x^6 + 15120*x^5 - 90720*x^4 + 423360*x^3 - 1451520*x^2 + 3265920*x - 362880*Sin[x] - 1814400)*Cos[x]^4 + ((- 9*x^8 + 144*x^7 - 1512*x^6 + 12096*x^5 - 75600*x^4 + 362880*x^3 - 1270080*x^2 + 2903040*x - 1451520)*Sin[x] - 1107*x^8 + 17712*x^7 - 137592*x^6 + 713664*x^5 - 2646000*x^4 + 6894720*x^3 - 11430720*x^2 + 8709120*x - 2903040)*Cos[x]^3 + ((- 251*x^9 + 4518*x^8 - 40392*x^7 + 247968*x^6 - 1134000*x^5 + 3900960*x^4 - 9737280*x^3 + 15966720*x^2 - 13063680*x + 4354560)*Sin[x] - 3653*x^9 + 65754*x^8 - 543240*x^7 + 2776032*x^6 - 9752400*x^5 + 24040800*x^4 - 40219200*x^3 + 42094080*x^2 - 26127360*x + 7257600)*Cos[x]^2 + ((9108*x^8 - 145728*x^7 + 1037232*x^6 - 4354560*x^5 + 11793600*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*Sin[x] + 13572*x^8 - 217152*x^7 + 1502928*x^6 - 5806080*x^5 + 13608000*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*Cos[x] + (11092*x^9 - 199656*x^8 + 1599696*x^7 - 7499520*x^6 + 22680000*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)*Sin[x] + 11588*x^9 - 208584*x^8 + 1666224*x^7 - 7741440*x^6 + 23042880*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)/((362880*Sin[x] - 1814400)*Cos[x]^4 + (- 4354560*Sin[x] + 7257600)*Cos[x]^2 + 5806080*Sin[x] - 5806080), {x, 0, nmax}], x] * Range[0, nmax]!, 10] (* Vaclav Kotesovec, Apr 30 2018 *)

Formula

a(n) ~ (4 - Pi)^9 * 2^(n + 5/2) * n^(n + 19/2) / (9! * Pi^(n + 19/2) * exp(n)). - Vaclav Kotesovec, Apr 29 2018
E.g.f.: (362880*cos(x)^5 + (x^9 - 18*x^8 + 216*x^7 - 2016*x^6 + 15120*x^5 - 90720*x^4 + 423360*x^3 - 1451520*x^2 + 3265920*x - 362880*sin(x) - 1814400)*cos(x)^4 + ((- 9*x^8 + 144*x^7 - 1512*x^6 + 12096*x^5 - 75600*x^4 + 362880*x^3 - 1270080*x^2 + 2903040*x - 1451520)*sin(x) - 1107*x^8 + 17712*x^7 - 137592*x^6 + 713664*x^5 - 2646000*x^4 + 6894720*x^3 - 11430720*x^2 + 8709120*x - 2903040)*cos(x)^3 + ((- 251*x^9 + 4518*x^8 - 40392*x^7 + 247968*x^6 - 1134000*x^5 + 3900960*x^4 - 9737280*x^3 + 15966720*x^2 - 13063680*x + 4354560)*sin(x) - 3653*x^9 + 65754*x^8 - 543240*x^7 + 2776032*x^6 - 9752400*x^5 + 24040800*x^4 - 40219200*x^3 + 42094080*x^2 - 26127360*x + 7257600)*cos(x)^2 + ((9108*x^8 - 145728*x^7 + 1037232*x^6 - 4354560*x^5 + 11793600*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*sin(x) + 13572*x^8 - 217152*x^7 + 1502928*x^6 - 5806080*x^5 + 13608000*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*cos(x) + (11092*x^9 - 199656*x^8 + 1599696*x^7 - 7499520*x^6 + 22680000*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)*sin(x) + 11588*x^9 - 208584*x^8 + 1666224*x^7 - 7741440*x^6 + 23042880*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)/((362880*sin(x) - 1814400)*cos(x)^4 + (- 4354560*sin(x) + 7257600)*cos(x)^2 + 5806080*sin(x) - 5806080). - Vaclav Kotesovec, Apr 30 2018

A343170 Triangle read by rows: coefficients of type B alternating Eulerian polynomials.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 11, 13, 13, 11, 57, 76, 118, 76, 57, 361, 597, 962, 962, 597, 361, 2763, 5270, 9733, 10548, 9733, 5270, 2763, 24611, 53849, 107427, 136673, 136673, 107427, 53849, 24611, 250737, 616408, 1334556, 1875432, 2167654, 1875432, 1334556, 616408, 250737
Offset: 0

Views

Author

N. J. A. Sloane, Apr 21 2021

Keywords

Examples

			Triangle begins:
   1;
   1,  1;
   3,  2,   3;
  11, 13,  13, 11;
  57, 76, 118, 76, 57;
  ...

Links

Crossrefs

Cf. A145876, A343171.

Programs

  • Mathematica
    Bhat[0][] = 1; Bhat[1][x] := 1 + x;
Bhat[n_][x_] := Bhat[n][x] = (n + x + (n-1) x^2) Bhat[n - 1][x] + (1 - x)* (1 + x^2) Bhat[n-1]'[x];
Table[CoefficientList[Bhat[n][x], x], {n, 0, 8}] // Flatten (* Jean-François Alcover, Apr 21 2021 *)

Extensions

More terms from Jean-François Alcover, Apr 21 2021
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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