A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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, 0, 34, 167, 339, 366, 225, 78, 14, 1, 0, 55, 310, 741, 976, 770, 370, 105, 16, 1, 0, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 0, 144, 1020, 3174, 5696, 6505, 4920, 2485, 824, 171, 20, 1

Offset

0,5

Comments

A Fibonacci convolution triangle; Riordan array (1, x*(1+x)/(1-x-x^2)).

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

Recurrence: T(n+2,k+1) = T(n+1,k+1) + T(n+1,k) + T(n,k+1) + T(n,k).

Explicit formula: T(n,k) = Sum_{i=0..floor(n/2)} binomial(n-i, i)*binomial(n-i, k)*k/(n-i), for n > 0.

G.f.: (1-x-x^2)/(1-(1+y)*x-(1+y)*x^2). - Philippe Deléham, Feb 21 2012

Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A155020(n), A154964(n), A154968(n), A154996(n), A154997(n), A154999(n), A155000(n), A155001(n), A155017(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Feb 21 2012

Sum_{k=0..n} T(n, k)*(m-1)^k = (1/m)*[n=0] - (m-1)*(i*sqrt(m))^(n-2)*ChebyshevU(n, -i*sqrt(m)/2). - G. C. Greubel, Mar 26 2021

Sum_{k=0..n} k * T(n,k) = A291385(n-1) for n>=1. - Alois P. Heinz, Sep 29 2022

Example

Triangle 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;
  ...

Maple

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n+1)); # Peter Luschny, Oct 19 2022

Mathematica

T[n_, k_]:= If[n==0, 1, Sum[Binomial[n-j, j]*Binomial[n-j, k]*k/(n-j), {j, 0, Floor[n/2]}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 26 2021 *)

Prog

(Magma)
T:= func< n, k | n eq 0 select 1 else (&+[ Binomial(n-j, j)*Binomial(n-j, k)*k/(n-j): j in [0..Floor(n/2)]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 26 2021
(Sage)
def T(n, k): return 1 if n==0 else sum( binomial(n-j, j)*binomial(n-j, k)*k/(n-j) for j in (0..n//2) )
flatten([[T(n, k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 26 2021

Crossrefs

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.

Cf. A000012, A155020, A154964, A154968, A154996, A154997, A154999, A155000, A155001, A155017.

Cf. A000045, A154929.

T(2n,n) gives A262910.

Cf. A291385.

Sequence in context: A128908 A285072 A300454 * A368099 A256130 A257566

Adjacent sequences: A155109 A155110 A155111 * A155113 A155114 A155115

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 20 2009

Extensions

Typos in two terms corrected by Alois P. Heinz, Aug 08 2015

Status

approved