A110441 Triangular array formed by the Mersenne numbers.

1, 3, 1, 7, 6, 1, 15, 23, 9, 1, 31, 72, 48, 12, 1, 63, 201, 198, 82, 15, 1, 127, 522, 699, 420, 125, 18, 1, 255, 1291, 2223, 1795, 765, 177, 21, 1, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1

Offset

0,2

Comments

This sequence factors A038255 into a product of Riordan arrays.

Subtriangle of the triangle given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012

From Peter Bala, Jul 22 2014: (Start)

Let M denote the lower unit triangular array A130330 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0 M/

having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

For 1<=k<=n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01 and 02. - Milan Janjic, Dec 20 2016

The convolution triangle of the Mersenne numbers. - Peter Luschny, Oct 09 2022

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)

Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, J. Int. Seq. 8 (2005), #05.3.7.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Formula

Riordan array M(n, k): (1/(1-3z+2z^2), z/(1-3z+2z^2)). Leftmost column M(n, 0) is the Mersenne numbers A000225, first column is A045618, second column is A055582, row sum is A007070 and diagonal sum is even-indexed Fibonacci numbers A001906.

T(n,k) = Sum_{j=0..n} C(j+k,k)C(n-j,k)2^(n-j-k). - Paul Barry, Feb 13 2006

From Philippe Deléham, Mar 19 2012: (Start)

G.f.: 1/(1-(3+y)*x+2*x^2).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) -2*T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000225(n+1), A007070(n), A107839(n), A154244(n), A186446(n), A190975(n+1), A190979(n+1), A190869(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7 respectively. (End)

Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (2^(i+1) - 1)*T(n-i,k). - Peter Bala, Jul 22 2014

From Peter Bala, Oct 07 2019: (Start)

Recurrence for row polynomials: R(n,x) = (3 + x)*R(n-1,x) - 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 3 + x.

The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 + 2*x/(1 + ... + x/(1 + 2*x/(1)))) (with 2*n partial numerators). Cf. A116414. (End)

Example

Triangle starts:
   1;
   3,  1;
   7,  6,  1;
  15, 23,  9,  1;
  31, 72, 48, 12,  1;
(0, 3, -2/3, 2/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1
  0,  1
  0,  3,  1
  0,  7,  6,  1
  0, 15, 23,  9,  1
  0, 31, 72, 48, 12, 1. - Philippe Deléham, Mar 19 2012
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
/ 1          \/1         \/1        \      / 1       \
| 3  1       ||0  1      ||0 1      |      | 3  1    |
| 7  3 1     ||0  3 1    ||0 0 1    |... = | 7  6 1  |
|15  7 3 1   ||0  7 3 1  ||0 0 3 1  |      |15 23 9 1|
|31 15 7 3 1 ||0 15 7 3 1||0 0 7 3 1|      |...      |
|...         ||...       ||...      |      |...      | - Peter Bala, Jul 22 2014

Maple

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022

Mathematica

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - (3 + y) x + 2 x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Crossrefs

Cf. A000225, A130330, A206306, A116414.

Sequence in context: A213043 A319740 A275662 * A111806 A372118 A321163

Adjacent sequences: A110438 A110439 A110440 * A110442 A110443 A110444

Keyword

easy,nonn,tabl

Author

Asamoah Nkwanta (nkwanta(AT)jewel.morgan.edu), Aug 08 2005

Status

approved