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.

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A258395 Number of 2n-length strings of balanced parentheses of exactly 7 different types that are introduced in ascending order.

Original entry on oeis.org

429, 40040, 2246244, 98760480, 3761539782, 130505896752, 4245988489600, 131928199603200, 3962683868528385, 116039722090972680, 3332921846278964940, 94315723869947580000, 2638390752595156276410, 73147630662437905413840, 2013841857892713303414960
Offset: 7

Views

Author

Alois P. Heinz, May 28 2015

Keywords

Links

Crossrefs

Column k=7 of A253180.

Formula

Recurrence: (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 56*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 1288*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 15680*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 108304*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 420224*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 836352*(n-5)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6) + 645120*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-7). - Vaclav Kotesovec, Jun 01 2015
a(n) ~ 28^n / (7!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

A258396 Number of 2n-length strings of balanced parentheses of exactly 8 different types that are introduced in ascending order.

Original entry on oeis.org

1430, 175032, 12597000, 698377680, 33079524324, 1411221754800, 55928745100800, 2100173331484800, 75727786603836510, 2646827388046104120, 90290940344491887000, 3021580012515765901200, 99583828881536195805180, 3242049884573075122369680
Offset: 8

Views

Author

Alois P. Heinz, May 28 2015

Keywords

Links

Crossrefs

Column k=8 of A253180.

Formula

Recurrence: (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 72*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 2184*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 36288*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 359184*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 2153088*(n-6)*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 7559936*(n-6)*(n-5)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6) + 14026752*(n-6)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-7) - 10321920*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-8). - Vaclav Kotesovec, Jun 01 2015
a(n) ~ 32^n / (8!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

A258397 Number of 2n-length strings of balanced parentheses of exactly 9 different types that are introduced in ascending order.

Original entry on oeis.org

4862, 755820, 67897830, 4633467300, 267074035800, 13733597077200, 650800305634050, 29021018652697500, 1235362166419751370, 50713478000403718500, 2022835296688063807950, 78843505678630977784500, 3016017325414346802772080, 113617986954086473298668800
Offset: 9

Views

Author

Alois P. Heinz, May 28 2015

Keywords

Links

Crossrefs

Column k=9 of A253180.

Formula

Recurrence: (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 90*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 3480*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 75600*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 1012368*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 8618400*(n-7)*(n-6)*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 46315520*(n-7)*(n-6)*(n-5)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6) + 150105600*(n-7)*(n-6)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-7) - 262803456*(n-7)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-8) + 185794560*(2*n - 17)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-9). - Vaclav Kotesovec, Jun 01 2015
a(n) ~ 36^n / (9!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

A258398 Number of 2n-length strings of balanced parentheses of exactly 10 different types that are introduced in ascending order.

Original entry on oeis.org

16796, 3233230, 354660460, 29214542500, 2013190058880, 122762429039250, 6850724997273300, 357603651626578500, 17726205673051976100, 843509478504416874150, 38843740303576863755100, 1741683387026398566250500, 76401095775145069217992560
Offset: 10

Views

Author

Alois P. Heinz, May 28 2015

Keywords

Comments

In general, column k>0 of A253180 is asymptotic to (4*k)^n / (k!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

Links

Crossrefs

Column k=10 of A253180.

Programs

  • Maple
    ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i/((k-i)!*i!), i=0..k):
a:= n-> T(n, 10):
seq(a(n), n=10..25);

Formula

Recurrence: (n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 110*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 5280*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 145200*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 2524368*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 28865760*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 218683520*(n-8)*(n-7)*(n-6)*(n-5)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6) + 1076416000*(n-8)*(n-7)*(n-6)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-7) - 3264915456*(n-8)*(n-7)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-8) + 5441863680*(n-8)*(2*n - 17)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-9) - 3715891200*(2*n - 19)*(2*n - 17)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-10). - Vaclav Kotesovec, Jun 01 2015
a(n) ~ 40^n / (10!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015
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