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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.)