A229001 -id:A229001 - cp's OEIS frontend

A228959 Total sum of squared lengths of ascending runs in all permutations of [n].

0, 1, 6, 32, 186, 1222, 9086, 75882, 705298, 7231862, 81160422, 990024466, 13047411482, 184788881838, 2799459801742, 45178128866282, 773829771302946, 14021761172671462, 267991492197471158, 5388234382450264002, 113692608262971520042, 2512031106415692960926

Offset: 0

Alois P. Heinz, Sep 09 2013

nonn

			a(0) = 0: ().
a(1) = 1: (1).
a(2) = 6 = 4+2: (1,2), (2,1).
a(3) = 32 = 9+5+5+5+5+3: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=2 of A229001.

a:= proc(n) option remember; `if`(n<3, n*(2*n-1),
      (2*n+1)*a(n-1) -(n-1)*((n+2)*a(n-2)-(n-2)*a(n-3)))
    end:
seq(a(n), n=0..30);

a[n_] := With[{k = 2}, Sum[If[n==t, 1, (n!/(t+1)!)(t(n-t+1)+1-((t+1)(n-t)+1)/(t+2))] t^k, {t, 1, n}]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A229001 *)

E.g.f.: (2*exp(x)-x-2)/(x-1)^2.
a(n) = (2*n+1)*a(n-1)-(n-1)*((n+2)*a(n-2)-(n-2)*a(n-3)) for n>=3, a(n) = n*(2*n-1) for n<3.
a(n) ~ n! * (2*exp(1)-3)*n. - Vaclav Kotesovec, Sep 12 2013

A228994 Total sum of the 4th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 18, 152, 1110, 8254, 66050, 578466, 5557246, 58381646, 667481754, 8262618730, 110195259446, 1576108225446, 24075493095346, 391282065519074, 6742907753730030, 122830141805635966, 2358555332361509066, 47617194132209848026, 1008436738991020480294

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=4 of A229001.

a:= proc(n) option remember; `if`(n<3, [0, 1, 18][n+1],
      ((12*n^3-42*n^2+38*n+7)*a(n-1)-(n-1)*(6*n^3-12*n^2+n+14)*a(n-2)
       +(n-1)*(n-2)*(6*n^2-12*n+7)*a(n-3))/(6*n^2-24*n+25))
    end:
seq(a(n), n=0..30);

a[n_] := With[{k = 4}, Sum[If[n==t, 1, (n!/(t+1)!)(t(n-t+1)+1-((t+1)(n-t)+1)/(t+2))] t^k, {t, 1, n}]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A229001 *)

E.g.f.: (exp(x)*(12*x^2-12*x+14)-x-14)/(x-1)^2.

a(n) ~ n! * (14*exp(1)-15)*n. - Vaclav Kotesovec, Sep 12 2013

A228995 Total sum of the 5th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 34, 378, 3186, 25620, 214410, 1930080, 18881250, 200907060, 2318843370, 28914797640, 387867845250, 5573855579340, 85476008430090, 1393770581296320, 24086416578328290, 439832565550644900, 8463528886854858090, 171191360282164168440, 3631513434281720800770

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=5 of A229001.

a:= proc(n) option remember; `if`(n<4, [0, 1, 34, 378][n+1],
      ((2*n^2-4*n+6)*a(n-1) -(n^3-3*n^2+13*n-10)*a(n-2)
      +2*(2*n-1)*(n-2)*a(n-3) +(n-3)*(n-2)^2*a(n-4)) /(n-2))
    end:
seq(a(n), n=0..30);

a[n_] := With[{k = 5}, Sum[If[n==t, 1, (n!/(t+1)!)(t(n-t+1)+1-((t+1)(n-t)+1)/(t+2))] t^k, {t, 1, n}]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A229001 *)

E.g.f.: (exp(x)*(20*x^3+30*x-30)+x+30)/(x-1)^2.

a(n) ~ n! * (20*exp(1)+31)*n. - Vaclav Kotesovec, Sep 12 2013

A228996 Total sum of the 6th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 66, 992, 9846, 86782, 765506, 7112202, 71000398, 766053422, 8931231882, 112221527986, 1514394506102, 21867699419238, 336675784490002, 5508056657818442, 95455624774115166, 1747299831395273182, 33693372749353108058, 682771622138237836962

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=6 of A229001.

a:= proc(n) option remember; `if`(n<3, [0, 1, 66][n+1],
      ((30*n^5-225*n^4+690*n^3-975*n^2+512*n+31)*a(n-1)
      -(n-1)*(15*n^5-90*n^4+255*n^3-330*n^2+121*n+62)*a(n-2)
      +(15*n^6-105*n^5+315*n^4-525*n^3+511*n^2-273*n+62)*a(n-3))/
      (15*n^4-120*n^3+375*n^2-540*n+301))
    end:
seq(a(n), n=0..30);

E.g.f.: (exp(x)*(30*x^4+60*x^3+60*x^2-60*x+62)-x-62)/(x-1)^2.

a(n) ~ n! * (152*exp(1)-63)*n. - Vaclav Kotesovec, Sep 12 2013

A228997 Total sum of the 7th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 130, 2706, 32226, 315684, 2961498, 28544040, 291590754, 3194874900, 37656861354, 477018980928, 6477756701010, 94006723773564, 1453236561824250, 23855684885059944, 414605141516228418, 7607828522859788580, 147012653519046471114, 2984603478905797978320

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=7 of A229001.

a:= proc(n) option remember; `if`(n<4, [0, 1, 130, 2706][n+1],
      ((16*n^3-38*n^2-16*n+278) *a(n-1)
      -(8*n^4-3*n^3-101*n^2+623*n-512) *a(n-2)
      +2*(n-2)*(8*n^3-32*n^2+134*n-95) *a(n-3)
      -(n-2)*(n-3)*(8*n^2-37*n+44) *a(n-4)) /(8*n^2-27*n+24))
    end:
seq(a(n), n=0..30);

With[{nn=20},CoefficientList[Series[(Exp[x](42x^5+210x^4+280x^3+126x-126)+x+126)/(x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 31 2021 *)

E.g.f.: (exp(x)*(42*x^5+210*x^4+280*x^3+126*x-126)+x+126)/(x-1)^2.

a(n) ~ n! * (532*exp(1)+127)*n. - Vaclav Kotesovec, Sep 12 2013

A228998 Total sum of the 8th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 258, 7592, 110310, 1217374, 12263090, 123349746, 1293790126, 14422297646, 172035525354, 2198386222330, 30052681253126, 438421632024006, 6806217982912546, 112117997189378354, 1954283594806071390, 35949546988844228446, 696172911589097791706

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Generally, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. Set k=8 for this sequence. - Vaclav Kotesovec, Sep 12 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=8 of A229001.

a:= proc(n) option remember; `if`(n<3, [0, 1, 258][n+1],
      ((56*n^7-644*n^6+3332*n^5-9590*n^4+16016*n^3-14588*n^2
       +5546*n+127)*a(n-1) -(n-1)*(28*n^7-280*n^6+1414*n^5
       -4060*n^4+6748*n^3-5992*n^2+2017*n+254)*a(n-2) +(n-1)*(n-2)*
      (28*n^6-168*n^5+490*n^4-840*n^3+868*n^2-504*n+127)*a(n-3))/
      (28*n^6-336*n^5+1750*n^4-5040*n^3+8428*n^2-7728*n+3025))
    end:
seq(a(n), n=0..30);

k=8; Table[n^k+Sum[t^k*n!*(n*(t^2+t-1)-t*(t^2-4)+1)/(t+2)!+Floor[t/n]*(1/(t*(t+3)+2)),{t,1,n-1}],{n,0,20}] (* Vaclav Kotesovec, Sep 12 2013 *)

a(n) ~ n! * (2914*exp(1)-255)*n. - Vaclav Kotesovec, Sep 12 2013

A228999 Total sum of the 9th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 514, 21738, 391026, 4924020, 53791050, 568417200, 6145964610, 69860663220, 844435826250, 10893660591480, 149983769015490, 2200169585018220, 34307707031519370, 567195605817520080, 9916409711089798530, 182880725414279970660, 3549284743350503607690

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Generally, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. Set k=9 for this sequence. - Vaclav Kotesovec, Sep 12 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=9 of A229001.

a:= proc(n) option remember; `if`(n<4, [0, 1, 514, 21738][n+1],
      ((462*n^4-1028*n^3+4428*n^2-18152*n+75890)*a(n-1)
      -(231*n^5-745*n^4+9964*n^3-46367*n^2+184023*n-145440)*a(n-2)
      +2*(n-2)*(2258*n^3-10187*n^2+41200*n-31605)*a(n-3)
      +(n-2)*(n-3)*(231*n^3-2258*n^2+6701*n-6340)*a(n-4))/
      (231*n^3-514*n^2-275*n+880))
    end:
seq(a(n), n=0..30);

k=9; Table[n^k+Sum[t^k*n!*(n*(t^2+t-1)-t*(t^2-4)+1)/(t+2)!+Floor[t/n]*(1/(t*(t+3)+2)),{t,1,n-1}],{n,0,20}] (* Vaclav Kotesovec, Sep 12 2013 *)

a(n) ~ n! * (14604*exp(1)+511)*n. - Vaclav Kotesovec, Sep 12 2013

A229000 Total sum of the 10th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 1026, 63152, 1424406, 20708542, 247753826, 2770103322, 31016696398, 360474871982, 4422094936842, 57643276901506, 799742156488022, 11800984833241638, 184874578304981362, 3068030670168269402, 53807082887654595486, 994936476288108004702

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Generally, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j) *C(k,j))) *exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. Set k=10 for this sequence. - Vaclav Kotesovec, Sep 12 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=10 of A229001.

a:= proc(n) option remember; `if`(n<5, [0, 1, 1026, 63152,
      1424406][n+1], ((398*n^3-539*n^2-4964*n+24377)*a(n-1)
      -(199*n^4+1057*n^3-12543*n^2+57436*n-31692)*a(n-2)
      +(1017*n^4-7565*n^3+34942*n^2-38827*n-17617)*a(n-3)
      -(n-3)*(1017*n^3-5258*n^2+21882*n-30370)*a(n-4)
      +(n-3)*(n-4)*(199*n^2-1212*n+1877)*a(n-5))/
      (199*n^2-778*n+792))
    end:
seq(a(n), n=0..30);

k=10; Table[n^k+Sum[t^k*n!*(n*(t^2+t-1)-t*(t^2-4)+1)/(t+2)!+Floor[t/n]*(1/(t*(t+3)+2)),{t,1,n-1}],{n,0,20}] (* Vaclav Kotesovec, Sep 12 2013 *)

a(n) ~ n! * (83342*exp(1)-1023)*n. - Vaclav Kotesovec, Sep 12 2013

A229003 Total sum of cubed lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 10, 66, 426, 2964, 22818, 195000, 1842234, 19120260, 216604194, 2662063728, 35297775930, 502460232684, 7644691295970, 123824090015544, 2127644969464698, 38659776477571860, 740692592536389474, 14924674961053224000, 315523813278300959994

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=3 of A229001.

a:= proc(n) option remember; `if`(n<3, [0, 1, 10][n+1],
      ((2*n^2-3*n-1)*a(n-1) -(n-1)*(n^2-2)*a(n-2)
       +(n-2)*(n-1)^2*a(n-3) )/(n-2))
    end:
seq(a(n), n=0..25);

With[{nn=20},CoefficientList[Series[(6 Exp[x](x-1)+x+6)/(x-1)^2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 26 2019 *)

E.g.f.: (6*exp(x)*(x-1)+x+6)/(x-1)^2.

a(n) ~ n! * 7*n. - Vaclav Kotesovec, Sep 12 2013

A229002 Total sum of the n-th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 6, 66, 1110, 25620, 765506, 28544040, 1293790126, 69860663220, 4422094936842, 323816329558128, 27127369640967206, 2575241880204602700, 274755427187762475922, 32708158728316937527944, 4316964980670466411606110, 628131523035069583394938980

Offset: 0

Alois P. Heinz, Sep 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Main diagonal of A229001.

a:= n-> add(`if`(n=k, 1, n!/(k+1)!*(k*(n-k+1)+1
        -((k+1)*(n-k)+1)/(k+2)))*k^n, k=1..n):
seq(a(n), n=1..20);

a(n) = Sum_{k=1..n} k^n * A122843(n,k).

cp's OEIS Frontend

A228959 Total sum of squared lengths of ascending runs in all permutations of [n].

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A228994 Total sum of the 4th powers of lengths of ascending runs in all permutations of [n].

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A228995 Total sum of the 5th powers of lengths of ascending runs in all permutations of [n].

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A228996 Total sum of the 6th powers of lengths of ascending runs in all permutations of [n].

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A228997 Total sum of the 7th powers of lengths of ascending runs in all permutations of [n].

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A228998 Total sum of the 8th powers of lengths of ascending runs in all permutations of [n].

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A228999 Total sum of the 9th powers of lengths of ascending runs in all permutations of [n].

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A229000 Total sum of the 10th powers of lengths of ascending runs in all permutations of [n].

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