A048487 -id:A048487 - cp's OEIS frontend

A250656 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

9, 16, 19, 25, 34, 39, 36, 53, 70, 79, 49, 76, 109, 142, 159, 64, 103, 156, 221, 286, 319, 81, 134, 211, 316, 445, 574, 639, 100, 169, 274, 427, 636, 893, 1150, 1279, 121, 208, 345, 554, 859, 1276, 1789, 2302, 2559, 144, 251, 424, 697, 1114, 1723, 2556, 3581

Offset: 1

R. H. Hardin, Nov 26 2014

Table starts
....9...16....25....36....49....64....81...100...121...144...169....196....225
...19...34....53....76...103...134...169...208...251...298...349....404....463
...39...70...109...156...211...274...345...424...511...606...709....820....939
...79..142...221...316...427...554...697...856..1031..1222..1429...1652...1891
..159..286...445...636...859..1114..1401..1720..2071..2454..2869...3316...3795
..319..574...893..1276..1723..2234..2809..3448..4151..4918..5749...6644...7603
..639.1150..1789..2556..3451..4474..5625..6904..8311..9846.11509..13300..15219
.1279.2302..3581..5116..6907..8954.11257.13816.16631.19702.23029..26612..30451
.2559.4606..7165.10236.13819.17914.22521.27640.33271.39414.46069..53236..60915
.5119.9214.14333.20476.27643.35834.45049.55288.66551.78838.92149.106484.121843

			Some solutions for n=4 k=4
..1..1..0..1..1....0..0..0..0..0....0..0..0..0..0....1..1..1..0..0
..0..0..0..1..1....1..1..1..1..1....1..1..1..1..1....0..0..0..0..0
..0..0..0..1..1....1..1..1..1..1....0..0..0..0..0....0..0..0..0..0
..0..0..0..1..1....0..0..0..0..0....1..1..1..1..1....1..1..1..1..1
..0..0..0..1..1....0..1..1..1..1....1..1..1..1..1....0..0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..880

Column 1 is A052549(n+1)
Column 2 is A176449
Column 3 is A156127(n+1)
Column 4 is A048487(n+2)
Row 1 is A000290(n+2)
Row 2 is A168244(n+3)

Empirical: T(n,k) = 2^(n-1)*k^2 + (5*2^(n-1)-1)*k + 2^(n+1)
Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1) +(5*2^(n-1) -1) +2^(n+1)
k=2: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*4 +(5*2^(n-1) -1)*2 +2^(n+1)
k=3: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*9 +(5*2^(n-1) -1)*3 +2^(n+1)
k=4: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*16 +(5*2^(n-1) -1)*4 +2^(n+1)
k=5: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*25 +(5*2^(n-1) -1)*5 +2^(n+1)
k=6: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*36 +(5*2^(n-1) -1)*6 +2^(n+1)
k=7: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*49 +(5*2^(n-1) -1)*7 +2^(n+1)
Empirical for row n:
n=1: a(n) = 1*n^2 + 4*n + 4
n=2: a(n) = 2*n^2 + 9*n + 8
n=3: a(n) = 4*n^2 + 19*n + 16
n=4: a(n) = 8*n^2 + 39*n + 32
n=5: a(n) = 16*n^2 + 79*n + 64
n=6: a(n) = 32*n^2 + 159*n + 128
n=7: a(n) = 64*n^2 + 319*n + 256

A048483 Array read by antidiagonals: T(k,n) = (k+1)2^n - k.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 10, 5, 1, 32, 31, 22, 13, 6, 1, 64, 63, 46, 29, 16, 7, 1, 128, 127, 94, 61, 36, 19, 8, 1, 256, 255, 190, 125, 76, 43, 22, 9, 1, 512, 511, 382, 253, 156, 91, 50, 25, 10, 1, 1024, 1023, 766, 509, 316, 187

Offset: 0

Clark Kimberling

n-th difference of (T(k,n),T(k,n-1),...,T(k,0)) is k+1, for n=1,2,3,...; k=0,1,2,...

			1 2 4 8 16 32 ...
1 3 7 15 31 63 ...
1 4 10 22 46 94 ...
1 5 13 29 61 125 ...
1 6 16 36 76 156 ...

Rows are A000079 (k=0), A000225 (k=1), A033484 (k=2), A036563 (k=3), A048487 (k=4), A048488 (k=5), A048489 (k=6), A048490 (k=7), A048491 (k=8).

Main diagonal is A048493. Cf. A048494.

G.f.: (1-x+kx)/[(1-x)(1-2x)]. E.g.f.: (k+1)*exp(2x) - k*exp(x).

Recurrences: T(k, n) = 2T(k, n-1)+k = T(k-1, n)+2^n-1, T(k, 0) = 1.

Edited by Ralf Stephan, Feb 05 2004

A051633 a(n) = 5*2^n - 2.

Original entry on oeis.org

3, 8, 18, 38, 78, 158, 318, 638, 1278, 2558, 5118, 10238, 20478, 40958, 81918, 163838, 327678, 655358, 1310718, 2621438, 5242878, 10485758, 20971518, 41943038, 83886078, 167772158, 335544318, 671088638, 1342177278, 2684354558, 5368709118, 10737418238, 21474836478

Offset: 0

Asher Auel

			a(5) = 5*2^4 - 2 = 80 - 2 = 78.

Stefano Spezia, Table of n, a(n) for n = 0..3300
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000079, A020714, A118654.

Cf. A123208, A048487, A131051, A153894.

LinearRecurrence[{3, -2},{3, 8},30] (* Ray Chandler, Jul 18 2020 *)

a(n) = A118654(n, 5).
a(n) = A000079(n)*5 - 2 = A020714(n) - 2. - Omar E. Pol, Dec 23 2008
a(n) = 2*(a(n-1)+1) with a(0)=3. - Vincenzo Librandi, Aug 06 2010
a(n) = A123208(2*n+1) = A048487(n)+2 = A131051(n+2) = A153894(n)-1. - Philippe Deléham, Apr 15 2013
G.f.: ( 3-x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Mar 23 2023
E.g.f.: exp(x)*(5*exp(x) - 2). - Stefano Spezia, Oct 03 2023

A134636 Triangle formed by Pascal's rule given borders = 2n+1.

Original entry on oeis.org

1, 3, 3, 5, 6, 5, 7, 11, 11, 7, 9, 18, 22, 18, 9, 11, 27, 40, 40, 27, 11, 13, 38, 67, 80, 67, 38, 13, 15, 51, 105, 147, 147, 105, 51, 15, 17, 66, 156, 252, 294, 252, 156, 66, 17, 19, 83, 222, 408, 546, 546, 408, 222, 83, 19, 21, 102, 305, 630, 954, 1092, 954, 630, 305, 102, 21

Offset: 0

Gary W. Adamson, Nov 04 2007

Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).

			First few rows of the triangle:
   1;
   3,  3;
   5,  6,  5;
   7, 11, 11,  7;
   9, 18, 22, 18,  9;
  11, 27, 40, 40, 27, 11;
  13, 38, 67, 80, 67, 38, 13;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A048487, A051601, A051597.

a134636 n k = a134636_tabl !! n !! k
a134636_row n = a134636_tabl !! n
a134636_tabl = iterate (\row -> zipWith (+) ([2] ++ row) (row ++ [2])) [1]
-- Reinhard Zumkeller, Nov 23 2012

T:= proc(n,k) option remember;
      `if`(k<0 or k>n, 0,
      `if`(k=0 or k=n, 2*n+1,
         T(n-1, k-1) + T(n-1, k) ))
    end:
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, May 26 2013

NestList[Append[Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] + 2], #[[1]] + 2] &, {1}, 10] // Grid  (* Geoffrey Critzer, May 26 2013 *)
T[n_, k_] := Binomial[n, k-1] + Binomial[n, k] + 2 Binomial[n, k+1] + Binomial[n, n-k+1];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

Triangle, given borders = (1, 3, 5, 7, 9, ...); apply Pascal's rule T(n,k) = T(n-1,k) P T(n-1,k-1).
T(n,k) = A051601(n,k) + A051597(n,k); T(n,k) mod 2 = A047999(n,k). - Reinhard Zumkeller, Nov 23 2012
Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

Offset changed by Reinhard Zumkeller, Nov 23 2012

A119726 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 4T(n-1, k-1) + 2T(n-1, k).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 16, 26, 1, 1, 36, 116, 106, 1, 1, 76, 376, 676, 426, 1, 1, 156, 1056, 2856, 3556, 1706, 1, 1, 316, 2736, 9936, 18536, 17636, 6826, 1, 1, 636, 6736, 30816, 76816, 109416, 84196, 27306, 1, 1, 1276, 16016, 88576, 276896, 526096, 606056, 391396, 109226, 1

Offset: 1

Zerinvary Lajos, Jun 14 2006

Second column is A048487.

Second diagonal is A020989.

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   16,    26,     1;
  1,   36,   116,   106,      1;
  1,   76,   376,   676,    426,      1;
  1,  156,  1056,  2856,   3556,   1706,      1;
  1,  316,  2736,  9936,  18536,  17636,   6826,      1;
  1,  636,  6736, 30816,  76816, 109416,  84196,  27306,      1;
  1, 1276, 16016, 88576, 276896, 526096, 606056, 391396, 109226, 1;

TERMESZET VILAGA XI.TERMESZET-TUDOMANY DIAKPALYAZAT 133.EVF. 6.SZ. jun. 2002. Vegh Lea (and Vegh Erika): "Pascal-tipusu haromszogek" http://www.kfki.hu/chemonet/TermVil/tv2002/tv0206/tartalom.html

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A007318, A020989, A048483, A048487, A119725, A119727, A123208.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 4*T(n-1,k-1) + 2*T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 18 2019

T:= proc(n, k) option remember;
      if k=1 and k=n then 1
    else 4*T(n-1, k-1) + 2*T(n-1, k)
      fi
end: seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 18 2019

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 4*T[n-1, k-1] + 2*T[n-1, k]]; Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==1 || k==n, 1, 4*T(n-1,k-1) + 2*T(n-1,k));

@CachedFunction
def T(n, k):
    if (k==1 or k==n): return 1
    else: return 4*T(n-1, k-1) + 2*T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 18 2019

Edited by Don Reble, Jul 24 2006

A123208 Start with 1, then alternately add 2 or double.

Original entry on oeis.org

1, 3, 6, 8, 16, 18, 36, 38, 76, 78, 156, 158, 316, 318, 636, 638, 1276, 1278, 2556, 2558, 5116, 5118, 10236, 10238, 20476, 20478, 40956, 40958, 81916, 81918, 163836, 163838, 327676, 327678, 655356, 655358, 1310716, 1310718, 2621436, 2621438, 5242876, 5242878

Offset: 0

Philippe Deléham, Oct 04 2006

			1, 1+2=3, 3*2=6, 6+2=8, 8*2=16, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A048487, A051633.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x+3*x^2-x^3)/((1-x^2)*(1-2*x^2)))); // Vincenzo Librandi, Jun 25 2013

a:=proc(n) if n mod 2 = 0 then 5*2^(n/2)-4 else 5*2^((n-1)/2)-2 fi end: seq(a(n),n=0..45); # Emeric Deutsch, Oct 10 2006

nxt[{a_,b_}]:={b+2,2(b+2)}; Rest[Flatten[NestList[nxt,{1,1},20]]] (* or *) LinearRecurrence[{0,3,0,-2},{1,3,6,8},40] (* Harvey P. Dale, Oct 10 2012 *)
CoefficientList[Series[(1 + 3 x + 3 x^2 - x^3) / ((1 - x) (1 + x) (1 - 2 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2013 *)

a(2n) = 5*2^n - 4; a(2n+1) = 5*2^n - 2 (n >= 0). - Emeric Deutsch, Oct 10 2006
From Colin Barker, Sep 10 2012: (Start)
a(n) = 3*a(n-2) - 2*a(n-4).
G.f.: (1+3*x+3*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)). (End)
a(2n) = A048487(n); a(2n+1) = A051633(n). - Philippe Deléham, Apr 15 2013
E.g.f.: 5*cosh(sqrt(2)*x) - 4*cosh(x) + 5*sinh(sqrt(2)*x)/sqrt(2) - 2*sinh(x). - Stefano Spezia, Oct 03 2023

More terms from Emeric Deutsch, Oct 10 2006

A131113 T(n,k) = 5binomial(n,k) - 4I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 5, 1, 5, 10, 1, 5, 15, 15, 1, 5, 20, 30, 20, 1, 5, 25, 50, 50, 25, 1, 5, 30, 75, 100, 75, 30, 1, 5, 35, 105, 175, 175, 105, 35, 1, 5, 40, 140, 280, 350, 280, 140, 40, 1, 5, 45, 180, 420, 630, 630, 420, 180, 45, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  5,  1;
  5, 10,  1;
  5, 15, 15,  1;
  5, 20, 30,  20,  1;
  5, 25, 50,  50, 25,  1;
  5, 30, 75, 100, 75, 30, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A048487, A131110, A131112, A131114, A131115.

T:= function(n,k)
    if k=n then return 1;
    else return 5*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

[k eq n select 1 else 5*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

seq(seq(`if`(k=n, 1, 5*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Table[If[k==n, 1, 5*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n, 1, 5*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

def T(n, k):
    if k == n: return 1
    else: return 5*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)]
# G. C. Greubel, Nov 18 2019

T(n,k) = 5*A007318(n,k) - 4*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 4*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021

A270810 Expansion of (x - x^2 + 2x^3 + 2x^4)/(1 - 3x + 2x^2).

Original entry on oeis.org

0, 1, 2, 6, 16, 36, 76, 156, 316, 636, 1276, 2556, 5116, 10236, 20476, 40956, 81916, 163836, 327676, 655356, 1310716, 2621436, 5242876, 10485756, 20971516, 41943036, 83886076, 167772156, 335544316, 671088636, 1342177276, 2684354556, 5368709116, 10737418236, 21474836476

Offset: 0

N. J. A. Sloane, Apr 06 2016

Colin Barker, Table of n, a(n) for n = 0..1000
M. Diepenbroek, M. Maus, A. Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Agrees with A048487 except for initial terms.
Cf. A002605, A265106, A265107, A265278.
Cf. A026646, A000225.

[n le 2 select n else 5*2^(n-2)-4: n in [0..40]]; // Bruno Berselli, Apr 08 2016

concat(0, Vec(x*(1-x+2*x^2+2*x^3)/((1-x)*(1-2*x)) + O(x^50))) \\ Colin Barker, Apr 12 2016

G.f.: x*(1 - x + 2*x^2 + 2*x^3)/((1 - x)*(1 - 2*x)).
a(n) = 5*2^(n-2)-4 for n>2. - Bruno Berselli, Apr 08 2016
a(n) = 3*a(n-1)-2*a(n-2) for n>4. - Colin Barker, Apr 12 2016
From Paul Curtz, Sep 23 2019: (Start)
a(n+1) = b(n+4) - b(n) where b(n) = 0, 1, 1, 1 followed by A026646.
a(n) = 2*a(n-1)+4 for n>4. (End)

A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 7, 4, 2, 1, 6, 10, 8, 4, 2, 1, 7, 13, 15, 8, 4, 2, 1, 8, 16, 22, 16, 8, 4, 2, 1, 9, 19, 29, 31, 16, 8, 4, 2, 1, 10, 22, 36, 46, 32, 16, 8, 4, 2, 1, 11, 25, 43, 61, 63, 32, 16, 8, 4, 2, 1, 12, 28, 50, 76, 94, 64, 32, 16, 8, 4, 2, 1, 13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1

Offset: 1

Henry Bottomley, May 29 2001

			Array begins as:
  1, 2, 3, 4,  5,  6,  7,   8,   9, ... A000027;
  1, 2, 4, 7, 10, 13, 16,  19,  22, ... A033627;
  1, 2, 4, 8, 15, 22, 29,  36,  43, ... A026474;
  1, 2, 4, 8, 16, 31, 46,  61,  76, ... A051039;
  1, 2, 4, 8, 16, 32, 63,  94, 125, ... A051040;
  1, 2, 4, 8, 16, 32, 64, 127, 190, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 255, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
Antidiagonal triangle begins as:
   1;
   2,  1;
   3,  2,  1;
   4,  4,  2,  1;
   5,  7,  4,  2,   1;
   6, 10,  8,  4,   2,   1;
   7, 13, 15,  8,   4,   2,  1;
   8, 16, 22, 16,   8,   4,  2,  1;
   9, 19, 29, 31,  16,   8,  4,  2,  1;
  10, 22, 36, 46,  32,  16,  8,  4,  2, 1;
  11, 25, 43, 61,  63,  32, 16,  8,  4, 2, 1;
  12, 28, 50, 76,  94,  64, 32, 16,  8, 4, 2, 1;
  13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Rows include A000027, A033627, A026474, A051039, A051040.
Diagonals include A000079, A000225, A033484, A036563, A048487.
Cf. A048488, A048489, A048490, A048491, A139634, A139635, A139697.
A048483 can be seen as half this table.

T[n_, k_]:= If[kG. C. Greubel, May 03 2022 *)

def A062001(n,k):
    if (kA062001(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, May 03 2022

If k <= n+1 then A(n, k) = 2^(k-1), while if k > n+1, A(n, k) = (2^n - 1)*(k - n) + 1 (array).
T(n, k) = A(k, n-k+1) (antidiagonals).
T(2*n-1, n) = A000079(n-1), n >= 1.
T(2*n, n) = A000079(n), n >= 1.
T(2*n+1, n) = A000225(n+1), n >= 1.
T(2*n+2, n) = A033484(n), n >= 1.
T(2*n+3, n) = A036563(n+3), n >= 1.
T(2*n+4, n) = A048487(n), n >= 1.
From G. C. Greubel, May 03 2022: (Start)
T(n, k) = (2^k - 1)*(n-2*k+1) + 1 for k < n/2, otherwise 2^(n-k).
T(2*n+5, n) = A048488(n), n >= 1.
T(2*n+6, n) = A048489(n), n >= 1.
T(2*n+7, n) = A048490(n), n >= 1.
T(2*n+8, n) = A048491(n), n >= 1.
T(2*n+9, n) = A139634(n), n >= 1.
T(2*n+10, n) = A139635(n), n >= 1.
T(2*n+11, n) = A139697(n), n >= 1. (End)

A204205 Triangle based on (0,1/5,1) averaging array.

Original entry on oeis.org

1, 1, 6, 1, 7, 16, 1, 8, 23, 36, 1, 9, 31, 59, 76, 1, 10, 40, 90, 135, 156, 1, 11, 50, 130, 225, 291, 316, 1, 12, 61, 180, 355, 516, 607, 636, 1, 13, 73, 241, 535, 871, 1123, 1243, 1276, 1, 14, 86, 314, 776, 1406, 1994, 2366, 2519, 2556, 1, 15, 100, 400

Offset: 1

Clark Kimberling, Jan 12 2012

See A204201 for a discussion and guide to other averaging arrays.

			First six rows:
1
1...6
1...7...16
1...8...23...36
1...9...31...59...76
1...10..40...90...135...156

Cf. A204201.

a = 0; r = 1/5; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]   (* A204205 triangle *)
Flatten[u]     (* A204205 sequence *)

T(n,n) = A048487(n-1). - Philippe Deléham, Dec 24 2013

T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=6, T(n,k)=0 if k<1 or if k>n. - Philippe Deléham, Dec 24 2013

cp's OEIS Frontend

A250656 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A048483 Array read by antidiagonals: T(k,n) = (k+1)2^n - k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A051633 a(n) = 5*2^n - 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A134636 Triangle formed by Pascal's rule given borders = 2n+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A119726 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 4*T(n-1, k-1) + 2*T(n-1, k).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Extensions

A123208 Start with 1, then alternately add 2 or double.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A131113 T(n,k) = 5*binomial(n,k) - 4*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A119726 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 4T(n-1, k-1) + 2T(n-1, k).

A131113 T(n,k) = 5binomial(n,k) - 4I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

A270810 Expansion of (x - x^2 + 2x^3 + 2x^4)/(1 - 3x + 2x^2).