A119258 -id:A119258 - cp's OEIS frontend

A118801 Triangle T that satisfies the matrix products: C[T^-1]C = T and T[C^-1]T = C, where C is Pascal's triangle.

1, 1, -1, 1, -3, 1, 1, -7, 5, -1, 1, -15, 17, -7, 1, 1, -31, 49, -31, 9, -1, 1, -63, 129, -111, 49, -11, 1, 1, -127, 321, -351, 209, -71, 13, -1, 1, -255, 769, -1023, 769, -351, 97, -15, 1, 1, -511, 1793, -2815, 2561, -1471, 545, -127, 17, -1, 1, -1023, 4097, -7423, 7937, -5503, 2561, -799, 161, -19, 1

Offset: 0

Paul D. Hanna, May 02 2006

Matrix inverse is triangle A118800. Row sums are: (1-n). Unsigned row sums equal A007051(n) = (3^n + 1)/2. Row squared sums equal A118802. Antidiagonal sums equal A080956(n) = (n+1)(2-n)/2. Unsigned antidiagonal sums form A024537 (with offset).
T = C^2*D^-1 where matrix product D = C^-1*T*C = T^-1*C^2 has only 2 nonzero diagonals: D(n,n)=-D(n+1,n)=(-1)^n, with zeros elsewhere. Also, [B^-1]*T*[B^-1] = B*[T^-1]*B forms a self-inverse matrix, where B^2 = C and B(n,k) = C(n,k)/2^(n-k). - Paul D. Hanna, May 04 2006
Riordan array ( 1/(1 - x), -x/(1 - 2*x) ) The matrix square is the Riordan array ( (1 - 2*x)/(1 - x)^2, x ), which belongs to the Appell subgroup of the Riordan group. See the Example section below. - Peter Bala, Jul 17 2013

			Formulas for initial columns are, for n>=0:
T(n+1,1) = 1 - 2^(n+1);
T(n+2,2) = 1 + 2^(n+1)*n;
T(n+3,3) = 1 - 2^(n+1)*(n*(n+1)/2 + 1);
T(n+4,4) = 1 + 2^(n+1)*(n*(n+1)*(n+2)/6 + n);
T(n+5,5) = 1 - 2^(n+1)*(n*(n+1)*(n+2)*(n+3)/24 + n*(n+1)/2 + 1).
Triangle begins:
1;
1,-1;
1,-3,1;
1,-7,5,-1;
1,-15,17,-7,1;
1,-31,49,-31,9,-1;
1,-63,129,-111,49,-11,1;
1,-127,321,-351,209,-71,13,-1;
1,-255,769,-1023,769,-351,97,-15,1;
1,-511,1793,-2815,2561,-1471,545,-127,17,-1;
1,-1023,4097,-7423,7937,-5503,2561,-799,161,-19,1; ...
The matrix square, T^2, starts:
1;
0,1;
-1,0,1;
-2,-1,0,1;
-3,-2,-1,0,1;
-4,-3,-2,-1,0,1; ...
where all columns are the same.
The matrix product C^-1*T*C = T^-1*C^2 is:
1;
-1,-1;
0, 1, 1;
0, 0,-1,-1;
0, 0, 0, 1, 1; ...
where C(n,k) = n!/(n-k)!/k!.

M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Q., 48 (2010), 290-297.

Cf. A118800 (inverse), A007051 (unsigned row sums), A118802 (Row squared sums), A080956 (antidiagonal sums), A024537 (unsigned antidiagonal sums).
A145661, A119258 and A118801 are all essentially the same (see the Shattuck and Waldhauser paper). - Tamas Waldhauser, Jul 25 2011
Cf. A007318, A032807, A097805, A112857, A130595, A156644, A167374, A200139, A239473.

Table[(1 + (-1)^k*2^(n - k + 1)*Sum[ Binomial[n - 2 j - 2, k - 2 j - 1], {j, 0, Floor[k/2]}]) - 4 Boole[And[n == 1, k == 0]], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

{T(n,k)=if(n==0&k==0,1,1+(-1)^k*2^(n-k+1)*sum(j=0,k\2,binomial(n-2*j-2,k-2*j-1)))}

T(n,k) = 1 + (-1)^k*2^(n-k+1)*Sum_{j=0..[k/2]} C(n-2j-2,k-2j-1) for n>=k>=0 with T(0,0) = 1.
For k>0, T(n,k) = -T(n-1,k-1) + 2*T(n-1,k). - Gerald McGarvey, Aug 05 2006
O.g.f.: (1 - 2*t)/(1 - t) * 1/(1 + t*(x - 2)) = 1 + (1 - x)*t + (1 - 3*x + x^2)*t^2 + (1 - 7*x + 5*x^2 - x^3)*t^3 + .... - Peter Bala, Jul 17 2013
From Tom Copeland, Nov 17 2016: (Start)
Let M = A200139^(-1) = (unsigned A118800)^(-1) and NpdP be the signed padded Pascal matrix defined in A097805. Then T(n,k) = (-1)^n* M(n,k) and T = P*NpdP = (A239473)^(-1)*P^(-1) = P*A167374*P^(-1) = A156644*P^(-1), where P is the Pascal matrix A007318 with inverse A130595. Cf. A112857.
Signed P^2 = signed A032807 = T*A167374. (End)

A027608 Expansion of 1/((1-x)(1-2x)^4).

Original entry on oeis.org

1, 9, 49, 209, 769, 2561, 7937, 23297, 65537, 178177, 471041, 1216513, 3080193, 7667713, 18808833, 45547521, 109051905, 258473985, 607125505, 1414529025, 3271557121, 7516192769, 17163091969

Offset: 0

N. J. A. Sloane

Michael De Vlieger, Table of n, a(n) for n = 0..3288
Michał Adamaszek and Henry Adams, On Vietoris-Rips complexes of hypercube graphs, arXiv:2103.01040 [math.CO], 2021.
M. H. Albert, M. D. Atkinson, and R. Brignall, The enumeration of three pattern classes using monotone grid classes, E. J. Combinat. 19 (3) (2012) P20. Chapter 5.5.1 (with leading zeros).
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
Santiago López de Medrano, On the genera of moment-angle manifolds associated to dual-neighborly polytopes, combinatorial formulas and sequences, arXiv:2003.07508 [math.GT], 2020.
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A001789 (first differences).

[(n/3)*(n^2+3*n+8)*2^n +1: n in [0..40]]; // G. C. Greubel, Aug 24 2022

CoefficientList[Series[1/((1-x)*(1-2x)^4), {x, 0, 22}], x] (* Michael De Vlieger, Jun 23 2020 *)
LinearRecurrence[{9,-32,56,-48,16},{1,9,49,209,769},30] (* Harvey P. Dale, Apr 09 2021 *)

Vec(1/((1-x)*(1-2*x)^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[(n/3)*(n^2+3*n+8)*2^n + 1 for n in (0..40)] # G. C. Greubel, Aug 24 2022

a(n-1) = 1 + (n-1)*2^(n+1) + ((n^3 - 7*n + 6)*2^(n-1))/3, n >= 1. - Roger Voles, Dec 07 2004, index corrected by R. J. Mathar, Mar 14 2011
a(n) = A119258(n+4,n). - Reinhard Zumkeller, May 11 2006
a(n) = 1 + n*2^(n+2) + (((n+1)^3 - 7*(n+1) + 6)*2^n)/3 = (n/3)*(n^2 + 3*n + 8)*2^n + 1, n >= 0. - Daniel Forgues, Nov 01 2012
E.g.f.: exp(x) + (8/3)*x*(3 + 3*x + x^2)*exp(2*x). - G. C. Greubel, Aug 24 2022

A193844 Triangular array: the fission of ((x+1)^n) by ((x+1)^n); i.e., the self-fission of Pascal's triangle.

Original entry on oeis.org

1, 1, 3, 1, 5, 7, 1, 7, 17, 15, 1, 9, 31, 49, 31, 1, 11, 49, 111, 129, 63, 1, 13, 71, 209, 351, 321, 127, 1, 15, 97, 351, 769, 1023, 769, 255, 1, 17, 127, 545, 1471, 2561, 2815, 1793, 511, 1, 19, 161, 799, 2561, 5503, 7937, 7423, 4097, 1023

Offset: 0

Clark Kimberling, Aug 07 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
A193844 is also the fission of (p1(n,x)) by (q1(n,x)), where p1(n,x)=x^n+x^(n-1)+...+x+1 and q1(n,x)=(x+2)^n.
Essentially A119258 but without the main diagonal. - Peter Bala, Jul 16 2013
From Robert Coquereaux, Oct 02 2014: (Start)
This is also a rectangular array A(n,p) read down the antidiagonals:
1 1 1 1 1 1 1 1 1
3 5 7 9 11 13 15 17 19
7 17 31 49 71 97 127 161 199
15 49 111 209 351 545 799 1121 1519
31 129 351 769 1471 2561 4159 6401 9439
...
Calling Gr(n) the Grassmann algebra with n generators, A(n,p) is the dimension of the space of Gr(n)-valued symmetric multilinear forms with vanishing graded divergence. If p is odd A(n,p) is the dimension of the cyclic cohomology group of order p of the Z2 graded algebra Gr(n). If p is even, the dimension of this cohomology group is A(n,p)+1. A(n,p) = 2^n*A059260(p,n-1)-(-1)^p.
(End)
The n-th row are also the coefficients of the polynomial P=sum_{k=0..n} (X+2)^k (in falling order, i.e., that of X^n first). - M. F. Hasler, Oct 15 2014

			First six rows:
1
1....3
1....5....7
1....7....17....15
1....9....31....49....31
1....11...49....111...129...63

Jean-François Chamayou, A Random Difference Equation with Dufresne Variables revisited, arXiv:1410.1708 [math.PR], 2014.
R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, J. of Geometry and Physics, 1995, Vol 15, pp 333-352.
R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, arXiv:hep-th/9310147, 1993.
C. Kassel, A Künneth formula for the cyclic cohomology of Z2-graded algebras, Math. Ann. 275 (1986) 683.

Cf. A193842, A193845, A119258.
Columns, diagonals: A000225, A000337, A055580, A027608, A211386, A211388, A000012, A005408, A056220, A199899.
A145661 is an essentially identical triangle.

A193844 := (n,k) -> 2^k*binomial(n+1,k)*hypergeom([1,-k],[-k+n+2],1/2);
for n from 0 to 5 do seq(round(evalf(A193844(n,k))),k=0..n) od; # Peter Luschny, Jul 23 2014
# Alternatively
p := (n,x) -> add(x^k*(1+2*x)^(n-k), k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n,x), x) od; # Peter Luschny, Jun 18 2017

z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (x + 1)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193844 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193845 *)

# uses[fission from A193842]
p = lambda n,x: (x+1)^n
A193844_row = lambda n: fission(p, p, n)
for n in range(7): print(A193844_row(n)) # Peter Luschny, Jul 23 2014

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} (-1)^i*binomial(n+1,k-i)*2^(k-i).
O.g.f.: 1/( (1 - x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + 3*x)*t + (1 + 5*x + 7*x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(x+1)*( (2*x+1)^(n+1) - x^(n+1) ). (End)
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014
T(n,k) = 2^k*binomial(n+1,k)*hyper2F1(1,-k,-k+n+2, 1/2). - Peter Luschny, Jul 23 2014

A135233 Triangle A007318 * A193554, read by rows.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 7, 5, 1, 41, 15, 17, 7, 1, 122, 31, 49, 31, 9, 1, 365, 63, 129, 111, 49, 11, 1, 1094, 127, 321, 351, 209, 71, 13, 1, 3281, 255, 769, 1023, 769, 351, 97, 15, 1, 9842, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = 3^n.

Left column = A007051: (1, 2, 5, 14, 41, 122, ...).

			First few rows of the triangle:
   1;
   2,  1;
   5,  3,  1;
  14,  7,  5,  1;
  41, 15, 17,  7,  1;
...

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

Cf. A007051, A007318, A118801, A119258, A193554.

function T(n,k)
  if k eq n then return 1;
  elif k eq 0 then return (3^n+1)/2;
  else return (&+[(-1)^(n-k+j)*2^j*Binomial(n, j): j in [0..n-k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then (3^n_1)/2
    else add((-1)^(n-k+j)*binomial(n, j)*2^j, j=0..n-k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3^n+1)/2, Sum [(-1)^(n-k+i)* Binomial[n, i]*2^i, {i, 0, n-k}]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==n, 1, if(k==0, (3^n+1)/2, sum(j=0, n-k, (-1)^(n-k+j)*binomial(n,j)*2^j) )); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==n): return 1
    elif (k==0): return (3^n+1)/2
    else: return sum((-1)^(n-k+j)*2^j*binomial(n, j) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

Binomial transform of A193554, as infinite lower triangular matrices.

T(n,k) = Sum_{j=0..n-k} (-1)^(n-k+j)*binomial(n,j)*2^j, with T(n,n) = 1, and T(n,0) = (3^n + 1)/2. - G. C. Greubel, Nov 20 2019

Definition corrected by N. J. A. Sloane, Jul 30 2011

A193823 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 9, 9, 1, 7, 19, 27, 27, 1, 9, 33, 65, 81, 81, 1, 11, 51, 131, 211, 243, 243, 1, 13, 73, 233, 473, 665, 729, 729, 1, 15, 99, 379, 939, 1611, 2059, 2187, 2187, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 6561, 6561, 1, 19, 163, 835, 2851

Offset: 0

Clark Kimberling, Aug 06 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

			First six rows:
1
1....1
1....3....3
1....5....9....9
1....7....19...27...27
1....9....33...65...81...81

Cf. A193722, A193804. A119258, A193860.

z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193823 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} binomial(n-1,k-i)*2^(k-i) for 0 <= k <= n.
O.g.f.: (1 - 2*x*t)^2/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + x)*t + (1 + 3*x + 3*x^2)*t^2 + .... Cf. A193860.
For n >= 1, the n-th row polynomial R(n,x) = 1/(x-1)*( 3^(n-1)*x^(n+1) - (2*x + 1)^(n-1) ). (End)

A119673 T(n, k) = 3*T(n-1, k-1) + T(n-1, k) for k < n and T(n, n) = 1, T(n, k) = 0, if k < 0 or k > n; triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 13, 1, 1, 10, 34, 40, 1, 1, 13, 64, 142, 121, 1, 1, 16, 103, 334, 547, 364, 1, 1, 19, 151, 643, 1549, 2005, 1093, 1, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 1, 28, 349, 2542, 11926, 37384, 78322, 105796, 83653, 29524, 1

Offset: 0

Zerinvary Lajos, Jun 11 2006

			Triangle begins:
  1;
  1,  1;
  1,  4,   1;
  1,  7,  13,    1;
  1, 10,  34,   40,    1;
  1, 13,  64,  142,  121,     1;
  1, 16, 103,  334,  547,   364,     1;
  1, 19, 151,  643, 1549,  2005,  1093,     1;
  1, 22, 208, 1096, 3478,  6652,  7108,  3280,    1;
  1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1;

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

Cf. A003462, A014915, A081271, A119258.

function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return 3*T(n-1,k-1) + T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019

T := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)* hypergeom([1,n+1],[k+2],m)/(k+1)!; A119673 := (n,k) -> T(n,k,3);
seq(print(seq(round(evalf(A119673(n,k))),k=0..n)),n=0..10); # Peter Luschny, Jul 25 2014

T[, 0]=1; T[n, n_]=1; T[n_, k_]/; 0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

T(n,k) = if(k<0 || k>n, 0, if(k==n, 1, 3*T(n-1, k-1) +T(n-1,k)));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 18 2019

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return 3*T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

T(n,k) = R(n,k,3) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k, k+1)* hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

Definition clarified by Philippe Deléham, Jun 13 2006

Entry revised by N. J. A. Sloane, Jun 19 2006

A368487 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^j * binomial(j+k-1,j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 10, 17, 4, 1, 1, 17, 64, 49, 5, 1, 1, 26, 177, 334, 129, 6, 1, 1, 37, 401, 1457, 1549, 321, 7, 1, 1, 50, 793, 4776, 10417, 6652, 769, 8, 1, 1, 65, 1422, 12889, 48526, 67761, 27064, 1793, 9, 1, 1, 82, 2369, 30234, 176185, 442276, 411825, 105796, 4097, 10, 1

Offset: 0

Seiichi Manyama, Dec 26 2023

			Square array begins:
  1, 1,   1,    1,     1,      1, ...
  1, 2,   5,   10,    17,     26, ...
  1, 3,  17,   64,   177,    401, ...
  1, 4,  49,  334,  1457,   4776, ...
  1, 5, 129, 1549, 10417,  48526, ...
  1, 6, 321, 6652, 67761, 442276, ...

Columns k=0..3 give A000012, A000027(n+1), A000337(n+1), A367591.
Main diagonal gives A368488.
Cf. A119258, A368486.

T(n, k) = sum(j=0, n, k^j*binomial(j+k-1, j));

G.f. of column k: 1/((1-x) * (1-k*x)^k).

A142595 Triangle T(n,k) = 2T(n-1, k-1) + 2T(n-1, k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 22, 40, 22, 1, 1, 46, 124, 124, 46, 1, 1, 94, 340, 496, 340, 94, 1, 1, 190, 868, 1672, 1672, 868, 190, 1, 1, 382, 2116, 5080, 6688, 5080, 2116, 382, 1, 1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1

Offset: 1

Roger L. Bagula, Sep 22 2008

This triangle is dominated by the Eulerian numbers A008292.

			Triangle begins as:
  1;
  1,   1;
  1,   4,    1;
  1,  10,   10,     1;
  1,  22,   40,    22,     1;
  1,  46,  124,   124,    46,     1;
  1,  94,  340,   496,   340,    94,     1;
  1, 190,  868,  1672,  1672,   868,   190,    1;
  1, 382, 2116,  5080,  6688,  5080,  2116,  382,   1;
  1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A008292, A047849 (row sums), A119258.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 2*(T(n-1, k-1) + T(n-1, k));
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 13 2021

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 2*(T[n-1, k-1] +T[n-1, k])];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)
a[0] = {1}; a[1] = {1, 1};
a[n_]:= a[n]= 2*Join[a[n-1], {-1/2}] + 2*Join[{-1/2}, a[n-1]];
Table[a[n], {n,0,10}]//Flatten (* Roger L. Bagula, Dec 09 2008 *)

@CachedFunction
def T(n,k): return 1 if k==1 or k==n else 2*(T(n-1, k-1) + T(n-1, k))
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2021

Sum_{k=0..n} T(n, k) = (4^(n-1) + 2)/3 = A047849(n-1).

Edited by N. J. A. Sloane, Dec 11 2008

A142596 Triangle T(n, k) = T(n-1, k-1) + 3T(n-1, k) + 2T(n-1, k-1), with T(n,1) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 66, 126, 66, 1, 1, 201, 576, 576, 201, 1, 1, 606, 2331, 3456, 2331, 606, 1, 1, 1821, 8811, 17361, 17361, 8811, 1821, 1, 1, 5466, 31896, 78516, 104166, 78516, 31896, 5466, 1, 1, 16401, 112086, 331236, 548046, 548046, 331236, 112086, 16401, 1

Offset: 1

Roger L. Bagula, Sep 22 2008

			The triangle begins as:
  1;
  1,     1;
  1,     6,      1;
  1,    21,     21,      1;
  1,    66,    126,     66,      1;
  1,   201,    576,    576,    201,      1;
  1,   606,   2331,   3456,   2331,    606,      1;
  1,  1821,   8811,  17361,  17361,   8811,   1821,      1;
  1,  5466,  31896,  78516, 104166,  78516,  31896,   5466,     1;
  1, 16401, 112086, 331236, 548046, 548046, 331236, 112086, 16401, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A008292, A047851, A060187, A119258.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 13 2021

T[n_, k_]:= T[n,k]= If[k==1 || k==n, 1, T[n-1, k-1] +3*T[n-1, k] +2*T[n-1, k-1]];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)

@CachedFunction
def T(n,k): return 1 if k==1 or k==n else T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2021

T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1.

Sum_{k=1..n} T(n, k) = (6^(n-1) + 4)/5 = A047851(n-1). - G. C. Greubel, Apr 13 2021

Edited by G. C. Greubel, Apr 13 2021

A142597 Triangle read by rows: t(n,k)=t(n - 1, k - 1) + 4* t(n - 1, k) + 3*t(n - 1, k - 1).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 148, 288, 148, 1, 1, 596, 1744, 1744, 596, 1, 1, 2388, 9360, 13952, 9360, 2388, 1, 1, 9556, 46992, 93248, 93248, 46992, 9556, 1, 1, 38228, 226192, 560960, 745984, 560960, 226192, 38228, 1, 1, 152916, 1057680, 3148608

Offset: 1

Roger L. Bagula, Sep 22 2008

Row sums are: {1, 2, 10, 74, 586, 4682, 37450, 299594, 2396746, 19173962, ...}.

			Triangle begins:
{1},
{1, 1},
{1, 8, 1},
{1, 36, 36, 1},
{1, 148, 288, 148, 1},
{1, 596, 1744, 1744, 596, 1},
{1, 2388, 9360, 13952, 9360, 2388, 1},
{1, 9556, 46992, 93248, 93248, 46992, 9556, 1},
{1, 38228, 226192, 560960, 745984, 560960, 226192, 38228, 1},
{1, 152916, 1057680, 3148608, 5227776, 5227776, 3148608, 1057680, 152916, 1}

Cf. A142175, A142467.

Cf. A008292, A119258, A060187, A142175, A142467.

A[n_, 1] := 1 A[n_, n_] := 1 A[n_, k_] := A[n - 1, k - 1] + 4* A[n - 1, k] + 3*A[n - 1, k - 1]; a = Table[A[n, k], {n, 10}, {k, n}]; Flatten[a]

Edited by N. J. A. Sloane, Dec 07 2008

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