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 A110441 Triangular array formed by the Mersenne numbers. 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 A321163 A054458 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

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Last modified April 1 09:11 EDT 2023. Contains 361685 sequences. (Running on oeis4.)