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A202390 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. 4
1, 1, 1, 1, 3, 2, 1, 6, 8, 3, 1, 10, 21, 17, 5, 1, 15, 45, 58, 35, 8, 1, 21, 85, 154, 144, 68, 13, 1, 28, 147, 350, 452, 330, 129, 21, 1, 36, 238, 714, 1195, 1198, 719, 239, 34, 1, 45, 366, 1344, 2799, 3611, 2959, 1506, 436, 55 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

A202390 is jointly generated with A208340 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=(x+1)*u(n-1,x)+(x+1)v(n-1,x).  The alternating row sums of A202390, and also A208340, are 0 except for the first one.  See the Mathematica section.  [From Clark Kimberling, Feb 27 2012]

LINKS

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

FORMULA

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n<k.

G.f.: (1-x)/(1-(2+y)*x+(1-y^2)*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A108411(n), A000007(n), A000012(n), A025192(n), A122558(n) for x = -2, -1, 0, 1, 2 respectively.

EXAMPLE

Triangle begins :

1

1, 1

1, 3, 2

1, 6, 8, 3

1, 10, 21, 17, 5

1, 15, 45, 58, 35, 8

1, 21, 85, 154, 144, 68, 13

1, 28, 147, 350, 452, 330, 129, 21

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

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

v[n_, x_] := (x + 1)*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[%]  (* A202390 *)

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

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

TableForm[cv]

Flatten[%]  (* A208340 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]  (*row sums*)

Table[u[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)

CROSSREFS

Cf. A000012, A000217, A051744, A000045, A123585, A208340.

Sequence in context: A139624 A132276 A257558 * A210858 A114586 A052174

Adjacent sequences:  A202387 A202388 A202389 * A202391 A202392 A202393

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Dec 18 2011

STATUS

approved

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Last modified October 23 14:11 EDT 2019. Contains 328345 sequences. (Running on oeis4.)