A242153 -id:A242153 - cp's OEIS frontend

A242160 Number of ascent sequences of length n with exactly seven flat steps.

1, 8, 72, 600, 5280, 48312, 465036, 4708704, 50160825, 561623920, 6600378928, 81297463104, 1047817553016, 14109488456400, 198192170408400, 2899804394683680, 44131025207930595, 697636040687261280, 11441167306954104500, 194421818718469399200

Offset: 8

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 8..140

Column k=7 of A242153.

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 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, -1], x, 7]; Table[a[n], {n, 8, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(23/2) / (7! * 6^5 * sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242161 Number of ascent sequences of length n with exactly eight flat steps.

Original entry on oeis.org

1, 9, 90, 825, 7920, 78507, 813813, 8828820, 100321650, 1193450830, 14850852588, 193081474872, 2619543882540, 37037407198050, 545028468623100, 8336937634715580, 132393075623791785, 2180112627147691500, 37183793747600839625, 656173638174834222300

Offset: 9

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 9..140

Column k=8 of A242153.

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 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, -1], x, 8]; Table[a[n], {n, 9, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(27/2) / (8! * 6^6 * sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242162 Number of ascent sequences of length n with exactly nine flat steps.

Original entry on oeis.org

1, 10, 110, 1100, 11440, 122122, 1356355, 15695680, 189496450, 2386901660, 31351799908, 429069944160, 6112269059260, 90535884261900, 1392850530925700, 22231833692574880, 367758543399421625, 6298103145093331000, 111551381242802518875, 2041429096543928691600

Offset: 10

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 10..140

Column k=9 of A242153.

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 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, -1], x, 9]; Table[a[n], {n, 10, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(31/2) / (9! * 6^7 * sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242163 Number of ascent sequences of length n with exactly ten flat steps.

Original entry on oeis.org

1, 11, 132, 1430, 16016, 183183, 2170168, 26682656, 341093610, 4535113154, 62703599816, 901046882736, 13446991930372, 208232533802370, 3342841274221680, 55579584231437200, 956172212838496225, 17004878491751993700, 312343867479847052850, 5920144379977393205640

Offset: 11

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 11..140

Column k=10 of A242153.

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 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, -1], x, 10]; Table[a[n], {n, 11, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(35/2) / (10! * 6^8 * sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242164 Number of ascent sequences of length 2n with exactly n flat steps.

Original entry on oeis.org

1, 1, 3, 20, 175, 2016, 28182, 465036, 8828820, 189496450, 4535113154, 119706872376, 3454013050488, 108140144894600, 3650830138093500, 132194177662402800, 5110163818369981650, 210037720563156731850, 9146299175093615073000, 420627290039763259876500

Offset: 0

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

			a(0) = 1: the empty sequence.
a(1) = 1: [0,0].
a(2) = 3: [0,0,0,1], [0,0,1,1], [0,1,1,1].
a(3) = 20: [0,0,0,0,1,0], [0,0,0,0,1,2], [0,0,0,1,0,0], [0,0,0,1,1,0], [0,0,0,1,1,2], [0,0,0,1,2,2], [0,0,1,0,0,0], [0,0,1,1,0,0], [0,0,1,1,1,0], [0,0,1,1,1,2], [0,0,1,1,2,2], [0,0,1,2,2,2], [0,1,0,0,0,0], [0,1,1,0,0,0], [0,1,1,1,0,0], [0,1,1,1,1,0], [0,1,1,1,1,2], [0,1,1,1,2,2], [0,1,1,2,2,2], [0,1,2,2,2,2].

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..70

Cf. A242153.

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 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[2n, -1, -1], x, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) = A242153(2n,n).

a(n) ~ 6*sqrt(3) / (Pi^3 * exp(Pi^2/12)) * (24/Pi^2)^n * n!. - Vaclav Kotesovec, Aug 28 2014

A193548 Decimal expansion of exp(Pi^2/12).

Original entry on oeis.org

2, 2, 7, 6, 1, 0, 8, 1, 5, 1, 6, 2, 5, 7, 3, 4, 0, 9, 4, 7, 9, 1, 0, 6, 1, 4, 1, 2, 0, 3, 1, 4, 9, 7, 4, 4, 6, 6, 9, 7, 9, 7, 4, 2, 6, 0, 3, 0, 0, 2, 3, 7, 7, 5, 6, 1, 5, 5, 1, 6, 1, 7, 0, 9, 8, 2, 7, 5, 0, 6, 3, 7, 2, 8, 6, 3, 0, 1, 4, 3, 1, 8, 6, 6, 8, 4, 6, 5, 7

Offset: 1

John M. Campbell, Jul 30 2011

Cf. A001113, A022493, A122214, A122215, A122216, A122217, A138265, A207651, A242153, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163, A242164.

Product[Product[k^((1/(n+1))*(-1)^(k)*Binomial[n,k-1]*HarmonicNumber[n]),{k,1,n+1}],{n,1,Infinity}]
RealDigits[E^(Pi^2/12), 10, 100]

exp(Pi^2/12) \\ Charles R Greathouse IV, Jul 30 2011

exp(Pi^2/12) = Product_{n>=1} Product_{k=1..n+1} k^(1/(n+1)) * H(n) * (-1)^k * binomial(n, k-1) where H(n) is the n-th harmonic number.

exp(Pi^2/12) = lim_{n -> infinity} Product_{k=1..n} (1 + k/n)^(1/k). - Peter Bala, Feb 14 2015

cp's OEIS Frontend

A242160 Number of ascent sequences of length n with exactly seven flat steps.

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A242161 Number of ascent sequences of length n with exactly eight flat steps.

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A242162 Number of ascent sequences of length n with exactly nine flat steps.

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A242163 Number of ascent sequences of length n with exactly ten flat steps.

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A242164 Number of ascent sequences of length 2n with exactly n flat steps.

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A193548 Decimal expansion of exp(Pi^2/12).

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