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 A185384 A binomial transform of Fibonacci numbers. 7
 1, 2, 1, 5, 6, 2, 13, 24, 15, 3, 34, 84, 78, 32, 5, 89, 275, 340, 210, 65, 8, 233, 864, 1335, 1100, 510, 126, 13, 610, 2639, 4893, 5040, 3115, 1155, 238, 21, 1597, 7896, 17080, 21112, 16310, 8064, 2492, 440, 34, 4181, 23256, 57492, 82908, 76860, 47502, 19572, 5184, 801, 55 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Triangle begins:      1,      2,    1,      5,    6,     2,     13,   24,    15,    3,     34,   84,    78,    32,     5,     89,  275,   340,   210,    65,    8,    233,  864,  1335,  1100,   510,  126,   13,    610, 2639,  4893,  5040,  3115, 1155,  238,  21,   1597, 7896, 17080, 21112, 16310, 8064, 2492, 440, 34,   ... Diagonal: a(n,n) = F(n+1). First column: a(n,0) = F(2n+1) (A001519). Row sums: Sum_{k=0..n} a(n,k) = F(3n+1) (A033887). Alternated row sums: Sum_{k=0..n} (-1)^k * a(n,k) = 1. Diagonal sums: Sum_{k=0..floor(n/2)} a(n-k,k) = A208481(n). Alternated diagonal sums: Sum_{k=0..floor(n/2)} (-1)^k * a(n-k,k) = F(n+3)-1 (A000071). Row square-sums: Sum_{k=0..n} a(n,k)^2 = A208588(n). Central coefficients: a(2*n,n) = binomial(2n,n)*F(3n+1) (A208473), where F(n) are the Fibonacci numbers (A000045). Mirror image of the triangle in A122070. - Philippe Deléham, Mar 13 2012 Subtriangle of (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 13 2012 LINKS G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened FORMULA a(n,k) = Sum_{i=k..n} binomial(n,i)*binomial(i,k)*F(i+1). a(n,k) = binomial(n,i) * Sum_{i=k..n} binomial(n-k,n-i)*F(i+1). Explicit form: a(n,k) = binomial(n,k)*F(2*n-k+1). G.f.: (1-x)/(1-3*x+x^2-x*y-x^2*y-x^2*y^2). Recurrence: a(n+2,k+2) = 3*a(n+1,k+2) + a(n+1,k+1) - a(n,k+2) + a(n,k+1) + a(n,k). T(n,k) = A122070(n,n-k). - Philippe Deléham, Mar 13 2012 EXAMPLE From Philippe Deléham, Mar 13 2012: (Start) (1, 1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:     1;     1,   0;     2,   1,    0;     5,   6,    2,    0;    13,  24,   15,    3,   0;    34,  84,   78,   32,   5,   0;    89, 275,  340,  210,  65,   8,  0;   233, 864, 1335, 1100, 510, 126, 13, 0;   ... (End) MATHEMATICA Flatten[Table[Sum[Binomial[n, i]Binomial[i, k]Fibonacci[i+1], {i, k, n}], {n, 0, 20}, {k, 0, n}]] CoefficientList[Series[CoefficientList[Series[(1 - x)/(1 - 3*x + x^2 - x*y - x^2*y - x^2*y^2), {x, 0, 10}], x], {y, 0, 10}], y] // Flatten (* G. C. Greubel, Jun 28 2017 *) PROG (Maxima) create_list(binomial(n, k)*fib(2*n-k+1), n, 0, 20, k, 0, n); (PARI) for(n=0, 10, for(k=0, n, print1(sum(i=k, n, binomial(n, i) * binomial(i, k) * fibonacci(i+1)), ", "))) \\ G. C. Greubel, Jun 28 2017 CROSSREFS Cf. A000045, A001519, A033887, A208481, A000071, A208588, A208473. Cf. A122070. Sequence in context: A231774 A209170 A231732 * A274728 A062991 A234950 Adjacent sequences:  A185381 A185382 A185383 * A185385 A185386 A185387 KEYWORD nonn,tabl,easy AUTHOR Emanuele Munarini, Feb 29 2012 STATUS approved

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Last modified September 21 07:28 EDT 2021. Contains 347596 sequences. (Running on oeis4.)