A010844 -id:A010844 - cp's OEIS frontend

A161380 Triangle read by rows: T(n,k) = 2kT(n-1,n-1) + 1 (n >= 0, 0 <= k <= n), with T(0,0) = 1.

1, 1, 3, 1, 7, 13, 1, 27, 53, 79, 1, 159, 317, 475, 633, 1, 1267, 2533, 3799, 5065, 6331, 1, 12663, 25325, 37987, 50649, 63311, 75973, 1, 151947, 303893, 455839, 607785, 759731, 911677, 1063623, 1, 2127247, 4254493, 6381739, 8508985, 10636231

Offset: 0

N. J. A. Sloane, Nov 28 2009

			Triangle begins:
1
1 3
1 7    13
1 27   53   79
1 159  317  475  633
1 1267 2533 3799 5065 6331

Nathaniel Johnston, Table of n, a(n) for n = 0..2500
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers. Discrete Math., 309 (2009), 6245-6254.

Diagonal gives A010844. Column 2 is A161370.

T := proc(n,k) option remember: if(n=0 and k=0)then return 1: else return 2*k*T(n-1,n-1)+1: fi: end:
for n from 0 to 8 do for k from 0 to n do printf("%d, ",T(n,k)): od: od: # Nathaniel Johnston, Apr 26 2011

T[0, 0] = 1; T[n_, k_] := 2*k*T[n - 1, n - 1] + 1;
Table[Table[T[n, k], {k, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)

A085644 a(0) = 1; a(n+1) = a(n)*2n + 2n + 1.

Original entry on oeis.org

1, 1, 5, 25, 157, 1265, 12661, 151945, 2127245, 34035937, 612646885, 12252937721, 269564629885, 6469551117265, 168208329048917, 4709833213369705, 141294996401091181, 4521439884834917825, 153728956084387206085, 5534242419037939419097, 210301211923441697925725

Offset: 0

Cino Hilliard, Aug 19 2003

Sum of reciprocals = 2.2472460058071954531775930749...

a(n) = 2 * A010844(n-1) - 1 = A007566(n-1) - 2n - 2 for n>=1.

a:= proc(n) option remember;
     `if`(n=0, 1, 2*((n-1)*a(n-1)+n)-1)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 14 2023

nxt[{n_,a_}]:={n+1,a*2n+2n+1}; Transpose[NestList[nxt,{1,1},20]][[2]] (* Harvey P. Dale, Aug 06 2016 *)

sum2x(n) = { s=1; sr=0; forstep(x=2,n,2, s=x*(s+1)+1; print1(s","); sr += 1.0/s; ); print(); print(sr) }

a(0)=1 prepended and edited by Alois P. Heinz, Mar 14 2023

A161381 Triangle read by rows: T(n,k) = n!*2^k/(n-k)! (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 4, 8, 1, 6, 24, 48, 1, 8, 48, 192, 384, 1, 10, 80, 480, 1920, 3840, 1, 12, 120, 960, 5760, 23040, 46080, 1, 14, 168, 1680, 13440, 80640, 322560, 645120, 1, 16, 224, 2688, 26880, 215040, 1290240, 5160960, 10321920, 1, 18, 288, 4032, 48384, 483840, 3870720, 23224320, 92897280, 185794560

Offset: 0

N. J. A. Sloane, Nov 28 2009

From Dennis P. Walsh, Nov 20 2012: (Start)
T(n,k) is the number of functions f:[k]->[2n] such that, if f(x)=f(y) or f(x)=2n+1-f(y), then x=y.
We call such functions injective-plus.
Equivalently, T(n,k) gives the number of ways to select k couples from n couples, then choose one person from each of the k selected couples, and then arrange those k individuals in a line. For example, T(50,10) is the number of ways to select 10 U.S. senators, one from each of ten different states, and arrange the senators in a reception line for a visiting dignitary. (End)

			Triangle begins:
  1
  1  2
  1  4  8
  1  6 24  48
  1  8 48 192  384
  1 10 80 480 1920 3840
For n=2 and k=2, T(2,2)=8 since there are exactly 8 functions f from {1,2} to {1,2,3,4} that are injective-plus. Letting f = <f(1),f(2)>, the 8 functions are <1,2>, <1,3>, <2,1>, <2,4>, <3,1>, <3,4>, <4,2>,and <4,3>. - _Dennis P. Walsh_, Nov 20 2012

Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, Discrete Math., 309 (2009), 6245-6254.
Dennis Walsh, Notes on injective-plus functions

A010844 (row sums). Cf. A008279.

/* As triangle */ [[Factorial(n)*2^k/Factorial((n-k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 23 2015

seq(seq(2^k*n!/(n-k)!,k=0..n),n=0..20); # Dennis P. Walsh, Nov 20 2012

Flatten@Table[Pochhammer[n - k + 1, k] 2^k, {n, 0, 20}, {k, 0, n}] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)

From Dennis P. Walsh, Nov 20 2012: (Start)
E.g.f. for column k: exp(x)*(2*x)^k.
G.f. for column k: (2*x)^k*k!/(1 - x)^(k+1).
T(n,k) = 2^(k-n)*Sum_{j = 0..n} (binomial(n,j)T(j,i)T(n-j,k-i). (End)
From Peter Bala, Feb 20 2016: (Start)
T(n, k) = 2*n*T(n-1, k-1) = 2*k*T(n-1, k-1) + T(n-1, k) = n*T(n-1, k)/(n - k) = 2*(n - k + 1)*T(n, k-1).
G.f. Sum_{n >= 1} (2*n*x*t)^(n-1)/(1 - (2*n*t - 1)*x)^n = 1 + (1 + 2*t)*x + (1 + 4*t + 8*t^2)*x^2 + ....
E.g.f. exp(x)/(1 - 2*x*t) = 1 + (1 + 2*t)*x + (1 + 4*t + 8*t^2)*x^2/2! + ....
E.g.f. for row n: (1 + 2*x)^n.
Row reversed triangle is the exponential Riordan array [1/(1 - 2*x), x]. (End)

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010

cp's OEIS Frontend

A161380 Triangle read by rows: T(n,k) = 2kT(n-1,n-1) + 1 (n >= 0, 0 <= k <= n), with T(0,0) = 1.

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A085644 a(0) = 1; a(n+1) = a(n)*2n + 2n + 1.

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A161381 Triangle read by rows: T(n,k) = n!*2^k/(n-k)! (n >= 0, 0 <= k <= n).

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cp's OEIS Frontend

A161380 Triangle read by rows: T(n,k) = 2*k*T(n-1,n-1) + 1 (n >= 0, 0 <= k <= n), with T(0,0) = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A085644 a(0) = 1; a(n+1) = a(n)*2n + 2n + 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A161381 Triangle read by rows: T(n,k) = n!*2^k/(n-k)! (n >= 0, 0 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A161380 Triangle read by rows: T(n,k) = 2kT(n-1,n-1) + 1 (n >= 0, 0 <= k <= n), with T(0,0) = 1.