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.

Showing 1-4 of 4 results.

A182542 Second diagonal of triangle in A145879.

Original entry on oeis.org

1, 8, 46, 232, 1093, 4944, 21778, 94184, 401930, 1698160, 7119516, 29666704, 123012781, 508019104, 2091005866, 8582372584, 35141476126, 143595498544, 585720020356, 2385430111024, 9701814930466, 39411044641888, 159926316674356, 648348726966672, 2626193752638388
Offset: 3

Views

Author

N. J. A. Sloane, May 04 2012

Keywords

Comments

Sum of valley heights over all Dyck n-paths. - David Scambler, Oct 05 2012

Examples

			Dyck 4-paths with nonzero valley heights are: UUUD(2)UDDD, UUUDD(1)UDD, UUD(1)UUDDD, UUD(1)UD(1)UDD, UUD(1)UDD(0)UD, and UD(0)UUD(1)UDD, with valley heights shown in parentheses, giving a(4) = 8. - _David Scambler_, Oct 05 2012

Links

Crossrefs

Cf. A145879.

Programs

  • Mathematica
    a[n_] := 4^(n - 1) - n CatalanNumber[n];
Array[a, 25, 3] (* Peter Luschny, Jun 08 2020 *)
  • Maxima
    a(n):=2*sum((4^i*binomial(2*(n-i),n-i-2))/(n-i),i,0,n-1); /* Vladimir Kruchinin, Mar 29 2019  */

Formula

G.f. appears to be (1-2*x-sqrt(1-4*x))^2/(4*x*(1-4*x)). - Mark van Hoeij, Apr 19 2013
a(n) ~ 2^(2*n-2) * (1-4/(sqrt(Pi*n))). - Vaclav Kotesovec, Aug 13 2013
a(n) = 2*Sum_{i=0..n-1} 4^i*C(2*(n-i),n-i-2)/(n-i). - Vladimir Kruchinin, Mar 29 2019
a(n) = 4^(n-1) - C(2*n,n)*n/(n+1). - Vladimir Kruchinin, Jun 08 2020

Extensions

More terms from Alois P. Heinz, May 30 2012

A182543 Penultimate diagonal of triangle in A145879.

Original entry on oeis.org

5, 8, 26, 112, 596, 3768, 27576, 229248, 2133792, 21983040, 248345280, 3052719360, 40563521280, 579385336320, 8852682585600, 144083913523200, 2488656760934400, 45465350973235200, 875935041046732800, 17749186274340864000, 377355425576693760000
Offset: 3

Views

Author

N. J. A. Sloane, May 04 2012

Keywords

Links

Crossrefs

Cf. A145879.

Programs

  • Mathematica
    Flatten[{5,8,26,RecurrenceTable[{-(-4+n)^2 (-3+n) a[-2+n]+(26-15 n+2 n^2) a[-1+n]+(5-n) a[n]==0,a[6]==112,a[7]==596},a,{n,6,25}]}] (* Vaclav Kotesovec, Sep 02 2014 *)

Formula

Recurrence (for n>=6): (n-5)*a(n) = (2*n^2 - 15*n + 26)*a(n-1) - (n-4)^2*(n-3)*a(n-2). - Vaclav Kotesovec, Sep 02 2014
a(n) ~ 2 * n! * (log(n) + gamma) / n^2, where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 02 2014

Extensions

More terms from Alois P. Heinz, May 30 2012

A289428 Sum of factorial-Catalan numbers over a certain family of decorations.

Original entry on oeis.org

1, 3, 18, 144, 1368, 14688, 173664, 2226528, 30647808
Offset: 0

Views

Author

N. J. A. Sloane, Jul 06 2017

Keywords

Comments

The precise definition involves the use of certain symbols akin to the peace symbol.

Links

  • Vincent Pilaud, V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016.

Crossrefs

Cf. A289427, A289429, A289430, A289431, A000142, A111537, A145879.

Formula

From Peter Bala, Dec 25 2019: (Start)
The following are conjectural (Cf. A145879):
Recurrence: a(n) = n*a(n-1) + 2*Sum_{k = 1..n} a(k-1)*a(n-k) with a(0) = 1.
O.g.f. as a regular C-fraction: 1/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - 5*x/(1 - 5*x/(1 - ... ))))))). Cf. A111537.
exp( Sum_{n >= 1} a(n)*(2*x)^n/n ) = 1 + 6*x + 54*x^2 + 636*x^3 + ... appears to have integer coefficients. (End)

A202992 Triangle T(n,k), read by rows, given by (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 5, 1, 0, 0, 14, 8, 2, 0, 0, 42, 46, 26, 6, 0, 0, 132, 232, 220, 112, 24, 0, 0, 429, 1093, 1527, 1275, 596, 120, 0, 0, 1430, 4944, 9436, 11384, 8638, 3768, 720, 0, 0, 4862, 21778, 54004, 87556, 95126, 66938, 27576, 5040, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 26 2011

Keywords

Comments

T(n,0) = A000108(n) (the Catalan numbers).
Row sums are the factorials (A000142).

Examples

			Triangle begins :
1
1, 0
2, 0, 0
5, 1, 0, 0
14, 8, 2, 0, 0
42, 46, 26, 6, 0, 0
132, 232, 220, 112, 24, 0, 0

Crossrefs

Cf. A000108, A000142, A011782, A145879

Formula

T(n,0) = A000108(n), Catalan numbers. Row sums = n! = A000142(n).
T(n+2,n) = n!, n>0.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A011782(n), A000108(n), A000142(n) for x = -1, 0, 1 respectively.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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