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A193845 Mirror of the triangle A193844. 4
1, 3, 1, 7, 5, 1, 15, 17, 7, 1, 31, 49, 31, 9, 1, 63, 129, 111, 49, 11, 1, 127, 321, 351, 209, 71, 13, 1, 255, 769, 1023, 769, 351, 97, 15, 1, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1, 1023, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This triangle is obtained by reversing the rows of the triangle A193844.

From Philippe Deléham, Jan 17 2014: (Start)

Subtriangle of the triangle in A112857.

T(n,0)   = A000225(n+1).

T(n,1)   = A000337(n).

T(n+2,2) = A055580(n).

T(n+3,3) = A027608(n).

T(n+4,4) = A211386(n).

T(n+5,5) = A211388(n).

T(n,n)   = A000012(n).

T(n+1,n) = A005408(n).

T(n+2,n) = A056220(n+2).

T(n+3,n) = A199899(n+1).

Row sums are A003462(n+1).

Diagonal sums are A048739(n).

Riordan array (1/((1-2*x)*(1-x), x/(1-2*x)). (End)

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-2)^0 + A_1*(x-2)^1 + A_2*(x-2)^2 + ... + A_n*(x-2)^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

The n-th row lists the coefficients of the polynomial sum_{k=0..n} (X+2)^k, in order of increasing powers. - M. F. Hasler, Oct 15 2014

LINKS

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

Russell Jay Hendel, A Method for Uniformly Proving a Family of Identities, arXiv:2107.03549 [math.CO], 2021.

FORMULA

T(n,k) = A193844(n,n-k).

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

EXAMPLE

First six rows:

1

3....1

7....5....1

15...17...7....1

31...49...31...9...1

63...129..111..49..11..1

MATHEMATICA

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 *)

PROG

(PARI) for(n=0, 20, for(k=0, n, print1(1/k!*sum(i=0, n, (2^(i-k)*prod(j=0, k-1, i-j))), ", "))) \\ Derek Orr, Oct 14 2014

CROSSREFS

Cf. A193844.

Cf. Columns: A000225, A000337, A055580, A027608, A211386, A211388

Cf. Diagonals: A000012, A005408, A056220, A199899

Sequence in context: A100584 A185877 A135858 * A265706 A098966 A021763

Adjacent sequences:  A193842 A193843 A193844 * A193846 A193847 A193848

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 07 2011

STATUS

approved

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Last modified October 19 09:35 EDT 2021. Contains 348074 sequences. (Running on oeis4.)