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 A154929 A Fibonacci convolution triangle. 11
 1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 8, 22, 21, 8, 1, 13, 45, 59, 36, 10, 1, 21, 88, 147, 124, 55, 12, 1, 34, 167, 339, 366, 225, 78, 14, 1, 55, 310, 741, 976, 770, 370, 105, 16, 1, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 144, 1020, 3174, 5696, 6505, 4920, 2485 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums are A028859. Diagonal sums are A141015(n+1). Inverse is A154930. Product of A030528 and A007318. Transforms sequence m^n with g.f. 1/(1-m*x) to the sequence with g.f. (1+x)/(1-(m+1)x-(m+1)x^2). Subtriangle of triangle T(n,k), given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. This triangle is the Riordan array (1, x(1+x)/(1-x-x^2)). - Philippe Deléham, Jan 25 2012 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150) Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4. FORMULA Riordan array ((1+x)/(1-x-x^2), x(1+x)/(1-x-x^2)); Triangle T(n,k) = Sum_{j=0..n} C(j+1,n-j)*C(j,k). T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(n,k)=0 if k > n. - Philippe Deléham, Jan 18 2009 Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A028859(n), A125145(n), A086347(n+1) for x=0,1,2,3 respectively. - Philippe Deléham, Jan 19 2009 EXAMPLE Triangle begins 1; 2, 1; 3, 4, 1; 5, 10, 6, 1; 8, 22, 21, 8, 1; 13, 45, 59, 36, 10, 1; 21, 88, 147, 124, 55, 12, 1; 34, 167, 339, 366, 225, 78, 14, 1; 55, 310, 741, 976, 770, 370, 105, 16, 1; Production array is 2, 1; -1, 2, 1; 3, -1, 2, 1; -10, 3, -1, 2, 1; 36, -10, 3, -1, 2, 1; -137, 36, -10, 3, -1, 2, 1; 543, -137, 36, -10, 3, -1, 2, 1; or ((1+x+sqrt(1+6x+5x^2))/2,x) beheaded. T(5,3) = T(4,3) + T(4,2) + T(3,3) + T(3,2) = 8 + 21 + 1 + 6 = 36. - Philippe Deléham, Jan 18 2009 From Philippe Deléham, Jan 25 2012: (Start) Triangle (0,2,-1/2,-1/2,0,0,0,...) DELTA (1,0,0,0,0,0,...) begins: 1; 0, 1; 0, 2, 1; 0, 3, 4, 1; 0, 5, 10, 6, 1; 0, 8, 22, 21, 8, 1; 0, 13, 45, 59, 36, 10, 1; 0, 21, 88, 147, 124, 55, 12, 1; (End) MATHEMATICA Table[Sum[Binomial[j + 1, n - j] Binomial[j, k], {j, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 25 2018 *) CROSSREFS Sequence in context: A094442 A060642 A306186 * A249042 A262472 A049400 Adjacent sequences: A154926 A154927 A154928 * A154930 A154931 A154932 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Jan 17 2009 STATUS approved

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Last modified March 28 14:02 EDT 2023. Contains 361595 sequences. (Running on oeis4.)