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A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k. 3
1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Rising and falling diagonals are A008999, A124400.

Subtriangle of triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2012

A102756 is jointly generated with A209130 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 05 2012

LINKS

Table of n, a(n) for n=0..54.

FORMULA

Sum_{k, 0<=k<=n}x^k*T(n,k) = A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for n= -1, 0, 1, 2, 3 respectively.

Sum_{k, 0<=k<=n}(-2)^k*T(n,k) = [1,0,3,0,9,0,27,0,81,0,...]for n=0,1,2,3,4,5,..., powers of 3 alternating with zeros.

T(n,n-1) = 2*A001629(n+1) for n>=1.

T(n,n) = Fibonacci(n+1) = A000045(n+1).

T(n,0) = n+1.

T(n,1) = A000292(n) for n>=1.

T(n+1,2) = binomial(n+4,n-1)+binomial(n+2,n-1)= A051747(n) for n>=1.

G.f.: 1/(1-(2+y)*x+(1+y)*(1-y)*x^2). - Philippe Deléham, Feb 17 2012

EXAMPLE

Triangle begins:

1;

2, 1;

3, 4, 2;

4, 10, 10, 3;

5, 20, 31, 20, 5;

6, 35, 76, 78, 40, 8;

7, 56, 161, 232, 184, 76, 13;

8, 84, 308, 582, 636, 406, 142, 21;

9, 120, 546, 1296, 1831, 1604, 861, 260, 34;

10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;

Triangle (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:

1

1, 0

2, 1, 0

3, 4, 2, 0

4, 10, 10, 3, 0

5, 20, 31, 20, 5, 0

6, 35, 76, 78, 40, 8, 0

7, 56, 161, 232, 184, 76, 13, 0

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];

v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]    (* A102756 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A209130 *)

(* Clark Kimberling, Mar 05 2012 *)

CROSSREFS

Cf. A209130.

Sequence in context: A120058 A208532 A245334 * A086614 A108959 A208750

Adjacent sequences:  A102753 A102754 A102755 * A102757 A102758 A102759

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Dec 18 2006

STATUS

approved

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Last modified October 23 08:57 EDT 2019. Contains 328345 sequences. (Running on oeis4.)