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A193843 Mirror image of the triangle A193842. 4
1, 4, 1, 13, 7, 1, 40, 34, 10, 1, 121, 142, 64, 13, 1, 364, 547, 334, 103, 16, 1, 1093, 2005, 1549, 643, 151, 19, 1, 3280, 7108, 6652, 3478, 1096, 208, 22, 1, 9841, 24604, 27064, 17086, 6766, 1720, 274, 25, 1, 29524, 83653, 105796, 78322, 37384, 11926 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A193843 is obtained by reversing the rows of the triangle A193842.

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-3)^0 + A_1*(x-3)^1 + A_2*(x-3)^2 + ... + A_n*(x-3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 14 2014

LINKS

Table of n, a(n) for n=0..50.

E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.

FORMULA

Write w(n,k) for the triangle at A193842. The triangle at A193843 is then given by w(n,n-k).

From Peter Bala, Jul 31 2012: (Start)

Matrix product of the shifted Pascal triangle {C(n+1,k+1)}n,k>=0 and the square of the Pascal triangle {2^(n-k)*C(n,k)}n,k>=0. Thus the triangle is the product of two triangular Galton arrays and so is also a Galton array (Neuwirth, Theorem 10).

T(n,k) = sum {i = 0..n} C(n+1,i+1)*2^(i-k)*C(i,k).

Riordan array [1/((1 - x)*(1 - 3*x)), x/(1 - 3*x)].

O.g.f.: 1/((1 - x)*(1 - (3+t)*x)) = 1 + (4+t)*x + (13+7*t+t^2)*x^2 + ....

First column A003462. Row sums A002450. Alternating row sums A000225. Antidiagonal sums (Sum_{k} T(n-k,k)) A082574. Weighted sums (Sum_{k} k*T(n,k)) A014916.

(End)

T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 4, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014

EXAMPLE

First six rows:

1;

4, 1;

13, 7, 1;

40, 34, 10, 1;

121, 142, 64, 13, 1;

364, 547, 334, 103, 16, 1;

MAPLE

T := proc(n, k) option remember;

if k<0 or k>n then 0 elif n=k then 1 elif n=1 and k=0 then 4

else 4*T(n-1, k) + T(n-1, k-1) -3*T(n-2, k) - T(n-2, k-1) fi end;

seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Jan 18 2014

MATHEMATICA

z = 10;

p[n_, x_] := (x + 1)^n;

q[n_, x_] := (x + 2)^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}]] (* A193842 *)

TableForm[Table[h[n], {n, 0, z}]] (* A193843 *)

Flatten[Table[h[n], {n, -1, z}]]

PROG

(PARI) for(n=0, 20, for(k=0, n, print1(1/k!*sum(i=0, n, (3^(i-k)*prod(j=0, k-1, i-j))), ", "))) \\ Derek Orr, Oct 14 2014

CROSSREFS

Cf. A193842.

Cf. A000225 (alt.row sums), A002450 (row sums), A014916 (weighted sums).

Cf. A003462 (first col.), A082574 (anti-diag.sums).

Sequence in context: A055252 A318945 A193956 * A116414 A215502 A144698

Adjacent sequences: A193840 A193841 A193842 * A193844 A193845 A193846

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 07 2011

STATUS

approved

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Last modified December 6 21:00 EST 2022. Contains 358648 sequences. (Running on oeis4.)