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%I #45 Oct 09 2022 08:21:11
%S 1,3,1,7,6,1,15,23,9,1,31,72,48,12,1,63,201,198,82,15,1,127,522,699,
%T 420,125,18,1,255,1291,2223,1795,765,177,21,1,511,3084,6562,6768,3840,
%U 1260,238,24,1,1023,7181,18324,23276,16758,7266,1932,308,27,1
%N Triangular array formed by the Mersenne numbers.
%C This sequence factors A038255 into a product of Riordan arrays.
%C 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
%C From _Peter Bala_, Jul 22 2014: (Start)
%C 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
%C /I_k 0\
%C \ 0 M/
%C 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)
%C 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
%C The convolution triangle of the Mersenne numbers. - _Peter Luschny_, Oct 09 2022
%H Michael De Vlieger, <a href="/A110441/b110441.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150)
%H Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, J. Int. Seq. 8 (2005), #05.3.7.
%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.
%F 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.
%F T(n,k) = Sum_{j=0..n} C(j+k,k)C(n-j,k)2^(n-j-k). - _Paul Barry_, Feb 13 2006
%F From _Philippe Deléham_, Mar 19 2012: (Start)
%F G.f.: 1/(1-(3+y)*x+2*x^2).
%F 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.
%F 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)
%F Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (2^(i+1) - 1)*T(n-i,k). - _Peter Bala_, Jul 22 2014
%F From _Peter Bala_, Oct 07 2019: (Start)
%F 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.
%F 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)
%e Triangle starts:
%e 1;
%e 3, 1;
%e 7, 6, 1;
%e 15, 23, 9, 1;
%e 31, 72, 48, 12, 1;
%e (0, 3, -2/3, 2/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
%e 1
%e 0, 1
%e 0, 3, 1
%e 0, 7, 6, 1
%e 0, 15, 23, 9, 1
%e 0, 31, 72, 48, 12, 1. - _Philippe Deléham_, Mar 19 2012
%e With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
%e / 1 \/1 \/1 \ / 1 \
%e | 3 1 ||0 1 ||0 1 | | 3 1 |
%e | 7 3 1 ||0 3 1 ||0 0 1 |... = | 7 6 1 |
%e |15 7 3 1 ||0 7 3 1 ||0 0 3 1 | |15 23 9 1|
%e |31 15 7 3 1 ||0 15 7 3 1||0 0 7 3 1| |... |
%e |... ||... ||... | |... | - _Peter Bala_, Jul 22 2014
%p # Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
%p PMatrix(10, n -> 2^n - 1); # _Peter Luschny_, Oct 09 2022
%t 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 *)
%Y Cf. A000225, A130330, A206306, A116414.
%K easy,nonn,tabl
%O 0,2
%A Asamoah Nkwanta (nkwanta(AT)jewel.morgan.edu), Aug 08 2005
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