A208588 -id:A208588 - cp's OEIS frontend

A185384 A binomial transform of Fibonacci numbers.

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

Offset: 0

Emanuele Munarini, Feb 29 2012

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

			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)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A000045, A001519, A033887, A208481, A000071, A208588, A208473.

Cf. A122070.

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 *)

create_list(binomial(n,k)*fib(2*n-k+1),n,0,20,k,0,n);

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

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

A122070 Triangle given by T(n,k) = Fibonacci(n+k+1)*binomial(n,k) for 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 2, 6, 5, 3, 15, 24, 13, 5, 32, 78, 84, 34, 8, 65, 210, 340, 275, 89, 13, 126, 510, 1100, 1335, 864, 233, 21, 238, 1155, 3115, 5040, 4893, 2639, 610, 34, 440, 2492, 8064, 16310, 21112, 17080, 7896, 1597, 55, 801, 5184, 19572, 47502, 76860, 82908, 57492, 23256, 4181

Offset: 0

Philippe Deléham, Oct 15 2006, Mar 13 2012

Subtriangle of (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Mirror image of the triangle in A185384.

			Triangle begins:
   1;
   1,   2;
   2,   6,   5;
   3,  15,  24,   13;
   5,  32,  78,   84,   34;
   8,  65, 210,  340,  275,  89;
  13, 126, 510, 1100, 1335, 864, 233;
(0, 1, 1, -1, 0, 0, ...) DELTA (1, 1, 1, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,   2;
  0,  2,   6,   5;
  0,  3,  15,  24,   13;
  0,  5,  32,  78,   84,   34;
  0,  8,  65, 210,  340,  275,  89;
  0, 13, 126, 510, 1100, 1335, 864, 233;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000045, A001519, A033887, A033889, A185384.

Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n+ k+1) ))); # G. C. Greubel, Oct 02 2019

[Binomial(n,k)*Fibonacci(n+k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 02 2019

with(combinat): seq(seq(binomial(n,k)*fibonacci(n+k+1), k=0..n), n=0..10); # G. C. Greubel, Oct 02 2019

Table[Fibonacci[n+k+1]*Binomial[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 02 2019 *)

T(n,k) = binomial(n,k)*fibonacci(n+k+1);
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 02 2019

[[binomial(n,k)*fibonacci(n+k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Oct 02 2019

T(n,k) = A000045(n+k+1)*A007318(n,k) .
T(n,n) = Fibonacci(2*n+1) = A001519(n+1) .
Sum_{k=0..n} T(n,k) = Fibonacci(3*n+1) = A033887(n) .
Sum_{k=0..n}(-1)^k*T(n,k) = (-1)^n = A033999(n) .
Sum_{k=0..floor(n/2)} T(n-k,k) = (Fibonacci(n+1))^2 = A007598(n+1).
Sum_{k=0..n} T(n,k)*2^k = Fibonacci(4*n+1) = A033889(n).
Sum_{k=0..n} T(n,k)^2 = A208588(n).
G.f.: (1-y*x)/(1-(1+3y)*x-(1+y-y^2)*x^2).
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A185384(n,n-k).
T(2n,n) = binomial(2n,n)*Fibonacci(3*n+1) = A208473(n).

Corrected and extended by Philippe Deléham, Mar 13 2012

Term a(50) corrected by G. C. Greubel, Oct 02 2019

cp's OEIS Frontend

A185384 A binomial transform of Fibonacci numbers.

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