A176305 -id:A176305 - cp's OEIS frontend

A193657 First difference of A002627.

1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Previous name was: Q-residue of the triangle A094727, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
Number of n X n rook placements avoiding the pattern 001. - N. J. A. Sloane, Feb 04 2013
Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 2, 3, etc., along the main diagonal, and zeros everywhere else. Then a(n) is equal to the permanent of M(n). - John M. Campbell, Apr 20 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013

Cf. A193649, A094727, A082425, A114870, A176305, A296964.

a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1,1):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 30 2014

q[n_, k_] := n + k + 1;  (* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193657 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
CoefficientList[Series[(E^x-x)/(x-1)^2,{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Nov 20 2012 *)

a(n) = { sum(k=0, n, if (k <= n-2, binomial(n,k)*(k+1)!, binomial(n,k)^2*k!));} \\ Michel Marcus, Feb 07 2013

def A193657():
    a = 2; b = 7; c = 31; n = 3
    yield 1
    while True:
        yield a
        n += 1
        a,b,c = b,c,((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
a = A193657(); [next(a) for n in range(19)] # Peter Luschny, May 30 2014

E.g.f.: (exp(x)-x)/(x-1)^2. - Vaclav Kotesovec, Nov 20 2012
a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012
a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - Peter Luschny, May 30 2014
a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 30 2014
From Peter Bala, Feb 10 2020: (Start)
a(n) = n*A002627(n) + 1.
a(n) = A114870(n) + n!.
a(n) = A296964(n+1) - A296964(n) for n >= 2.
a(1) = 2 and a(n) = (n^2*a(n-1) - 1)/(n - 1) for n >= 2. See A082425 for solutions to this recurrence with different starting values.
Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
Second column of A176305. (End)

Simpler definition by Peter Luschny, May 30 2014

A156072 T(n,k) = 1 + a(n) - a(k) - a(n - k), where a(n) = A078012(n+2), triangle read by rows (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 3, 5, 6, 6, 6, 5, 3, 1, 1, 4, 7, 9, 9, 9, 9, 7, 4, 1, 1, 6, 10, 13, 14, 14, 14, 13, 10, 6, 1, 1, 9, 15, 19, 21, 22, 22, 21, 19, 15, 9, 1, 1, 13, 22, 28, 31, 33, 34, 33, 31, 28, 22, 13, 1

Offset: 0

Roger L. Bagula, Feb 03 2009

			Triangle begins:
  1;
  1, 1;
  1, 0,  1;
  1, 0,  0,  1;
  1, 1,  1,  1,  1;
  1, 1,  2,  2,  1,  1;
  1, 1,  2,  3,  2,  1,  1;
  1, 2,  3,  4,  4,  3,  2,  1;
  1, 3,  5,  6,  6,  6,  5,  3,  1;
  1, 4,  7,  9,  9,  9,  9,  7,  4, 1;
  1, 6, 10, 13, 14, 14, 14, 13, 10, 6, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000930, A156070, A176305, A176306, A176307, A176625, A176339, A176344.

Clear[a, t, n, m];
a[0] = 0; a[1] = 1; a[2] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 3];
t[n_, m_] := 1 + a[n] - a[m] - a[n - m];
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

(a[0] : 0, a[1] : 1, a[2] : 1, a[n] := a[n - 1] + a[n - 3], T(n,k) := 1 + a[n] - a[k] - a[n-k])$ create_list(T(n, k), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Nov 25 2018 */

Edited and name clarified by Franck Maminirina Ramaharo, Nov 25 2018

A176344 T(n,k) = 1 + A176343(n) - A176343(k) - A176343(n-k), triangle read by rows (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 11, 13, 11, 1, 1, 65, 75, 75, 65, 1, 1, 568, 632, 640, 632, 568, 1, 1, 7789, 8356, 8418, 8418, 8356, 7789, 1, 1, 168761, 176549, 177114, 177168, 177114, 176549, 168761, 1, 1, 5847568, 6016328, 6024114, 6024671, 6024671, 6024114, 6016328, 5847568, 1

Offset: 0

Roger L. Bagula, Apr 15 2010

			Triangle begins:
  1;
  1,      1;
  1,      1,      1;
  1,      3,      3,      1;
  1,     11,     13,     11,      1;
  1,     65,     75,     75,     65,      1;
  1,    568,    632,    640,    632,    568,      1;
  1,   7789,   8356,   8418,   8418,   8356,   7789,      1;
  1, 168761, 176549, 177114, 177168, 177114, 176549, 168761, 1;
  ...

G. C. Greubel, Rows n = 0..75 of triangle, flattened

Cf. A156070, A156072, A176305, A176306, A176307, A176625, A176339.

b:= function(n)
    if n=0 then return 0;
    else return 1 + Fibonacci(n)*b(n-1);
    fi; end;
T:= function(n,k) return 1 + b(n) - b(n-k) - b(k); end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Dec 07 2019

function b(n)
  if n eq 0 then return 0;
  else return 1 + Fibonacci(n)*b(n-1);
  end if; return b; end function;
function T(n,k) return 1 + b(n) - b(n-k) - b(k); end function; [ T(n,k) : k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 07 2019

with(combinat);
b:= proc(n) option remember;
   if n = 0 then 0    else 1+fibonacci(n)*b(n-1)
   fi; end proc;
T:= proc (n, k) 1+b(n)-b(n-k)-b(k) end proc;
seq(seq(T(n, k), k = 0..n), n = 0..10); # G. C. Greubel, Dec 08 2019

b[n_]:= b[n]= If[n==0, 0, Fibonacci[n]*b[n-1] + 1]; (* A176343 *)
T[n_, k_]:= T[n, k] = 1 + a[n] - a[n-k] - a[k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Dec 08 2019 *)

(a[0] : 0, a[n] := fib(n)*a[n-1] + 1, T(n, m) := 1 + a[n] - a[m] - a[n-m])$ create_list(T(n, m), n, 0, 10, m, 0, n); /* Franck Maminirina Ramaharo, Nov 25 2018 */

b(n) = if(n==0, 0, 1 + fibonacci(n)*b(n-1) );
T(n,k) = 1 + b(n) - b(n-k) - b(k);
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 07 2019

def b(n):
    if (n==0): return 0
    else: return 1 + fibonacci(n)*b(n-1)
def T(n,k): return 1 + b(n) - b(n-k) - b(k)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 07 2019

Edited and name clarified by Franck Maminirina Ramaharo, Nov 25 2018

A176625 T(n,k) = 1 + 3k(k - n), triangle read by rows (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, -5, -5, 1, 1, -8, -11, -8, 1, 1, -11, -17, -17, -11, 1, 1, -14, -23, -26, -23, -14, 1, 1, -17, -29, -35, -35, -29, -17, 1, 1, -20, -35, -44, -47, -44, -35, -20, 1, 1, -23, -41, -53, -59, -59, -53, -41, -23, 1, 1, -26, -47, -62, -71, -74, -71, -62, -47

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins:
  1;
  1,   1;
  1,  -2,   1:
  1,  -5,  -5,   1;
  1,  -8, -11,  -8,   1;
  1, -11, -17, -17, -11,   1;
  1, -14, -23, -26, -23, -14,   1;
  1, -17, -29, -35, -35, -29, -17,   1;
  1, -20, -35, -44, -47, -44, -35, -20,   1;
  1, -23, -41, -53, -59, -59, -53, -41, -23,  1;
  1, -26, -47, -62, -71, -74, -71, -62, -47, -26, 1;
  ...

Cf. A000326, A156070, A156072, A176305, A176306, A176307, A176339, A176344.

/ * As triangle */ [[1 + 3*k*(k - n): k in [0..n]]: n in [0.. 15]];  // Vincenzo Librandi, Nov 26 2018

a[n_] = n*(3*n - 1)/2; (* A000326 *)
t[n_, m_] = 1 - a[n] + a[m] + a[n - m];
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

create_list(1 + 3*k*(k - n), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Nov 25 2018 */

T(n,k) = 1 - A000326(n) + A000326(k) + A000326(n-k).

Edited and name clarified by Franck Maminirina Ramaharo, Nov 25 2018

cp's OEIS Frontend

A193657 First difference of A002627.

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A156072 T(n,k) = 1 + a(n) - a(k) - a(n - k), where a(n) = A078012(n+2), triangle read by rows (n >= 0, 0 <= k <= n).

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A176344 T(n,k) = 1 + A176343(n) - A176343(k) - A176343(n-k), triangle read by rows (n >= 0, 0 <= k <= n).

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GAP

Magma

Maple

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PARI

Sage

Extensions

A176625 T(n,k) = 1 + 3*k*(k - n), triangle read by rows (n >= 0, 0 <= k <= n).

Views

Author

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Examples

Crossrefs

Programs

Magma

Mathematica

Maxima

Formula

Extensions

A176625 T(n,k) = 1 + 3k(k - n), triangle read by rows (n >= 0, 0 <= k <= n).