A052991 -id:A052991 - cp's OEIS frontend

A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 3, 1, 0, 5, 10, 9, 4, 1, 0, 8, 20, 22, 14, 5, 1, 0, 13, 38, 51, 40, 20, 6, 1, 0, 21, 71, 111, 105, 65, 27, 7, 1, 0, 34, 130, 233, 256, 190, 98, 35, 8, 1, 0, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 0, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1

Offset: 0

Philippe Deléham, Jan 21 2009

			Triangle begins:
[0] 1;
[1] 0,  1;
[2] 0,  1,   1;
[3] 0,  2,   2,   1;
[4] 0,  3,   5,   3,   1;
[5] 0,  5,  10,   9,   4,   1;
[6] 0,  8,  20,  22,  14,   5,  1;
[7] 0, 13,  38,  51,  40,  20,  6,  1;
[8] 0, 21,  71, 111, 105,  65, 27,  7, 1;
[9] 0, 34, 130, 233, 256, 190, 98, 35, 8, 1.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Row sums are in A215928.
Central terms: T(2*n,n) = A213684(n) for n > 0.
Cf. A000045, A037027, A122542, A059283, A321620.

a155161 n k = a155161_tabl !! n !! k
a155161_row n = a155161_tabl !! n
a155161_tabl = [1] : [0,1] : f [0] [0,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 17 2013

T := (n, k) -> binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..11); # Peter Luschny, May 23 2021
# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

CoefficientList[#, y]& /@ CoefficientList[(1-x-x^2)/(1-x-x^2-x*y)+O[x]^12, x] // Flatten (* Jean-François Alcover, Mar 01 2019 *)
(* Generates the triangle without the leading '1' (rows are rearranged). *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[x/(1 - x - x^2), 11] // Flatten  (* Peter Luschny, Feb 27 2021 *)

M(n,k):=pochhammer(n,k)/k!;
create_list(sum(M(k,i)*binomial(i,n-i-k),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 15 2011 */

T(n, k) given by [0,1,1,-1,0,0,0,...] DELTA [1,0,0,0,...] where DELTA is the operator defined in A084938.
a(n,k) = Sum_{i=0..n-k} M(k,i)*binomial(i,n-i-k), where M(n,k) = n(n+1)(n+2)...(n+k-1)/k!. - Emanuele Munarini, Mar 15 2011
Recurrence: a(n+2,k+1) = a(n+1,k+1) + a(n+1,k) + a(n,k+1). - Emanuele Munarini, Mar 15 2011
G.f.: (1-x-x^2)/(1-x-x^2-x*y). - Philippe Deléham, Feb 08 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n) (n > 0), A052991(n), A155179(n), A155181(n), A155195(n), A155196(n), A155197(n), A155198(n), A155199(n) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Feb 08 2012
T(n, k) = binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4). - Peter Luschny, May 23 2021

A181870 a(1) = 2, a(2) = 1. For n >= 3, a(n) is found by concatenating the cubes of the first n-1 terms of the sequence in reverse order and then dividing the resulting number by a(n-1).

Original entry on oeis.org

2, 1, 18, 32401, 1049824801000018, 110213211279472739469283600032400000000000000032401

Offset: 1

Peter Bala, Dec 09 2010

The calculations for the first few values of the sequence are
... 2^3 = 8 so a(3) = 18/1 = 18
... 18^3 = 5832 so a(4) = 583218/18 = 32401
... 32401^3 = 34015373377201 so a(5) = 34015373377201583218/32401 = 1049824801000018.
The value of a(7) is given in the Example section below. For similarly defined sequences see A181754 through A181756 and A181864 through A181869.

			a(7) = 12 14695 19405 33697 08585 73749 64887 08343 06977 30753 71161 37983 82152 46308 85158 49299 58480 00000 00001 04982 48010 00000 00000 00000 00000 00000 00000 00000 00000 00000 00001 04982 48010 00018 has 167 digits.

Cf. A006190, A052991, A181754, A181755, A181756, A181864, A181865, A181866, A181867, A181868, A181869

#A181870
M:=6: a:=array(1..M):s:=array(1..M):
a[1]:=2: a[2]:=1:
s[1]:=convert(a[1]^3,string): s[2]:=cat(convert(a[2]^3,string),s[1]):
for n from 3 to M do
a[n] := parse(s[n-1])/a[n-1];
s[n]:= cat(convert(a[n]^3,string),s[n-1]);
end do:
seq(a[n],n = 1..M);

DEFINITION
a(1) = 2, a(2) = 1, and for n >= 3,
(1)... a(n) = concatenate (a(n-1)^3,a(n-2)^3,...,a(1)^3)/a(n-1).
RECURRENCE RELATION
It appears that for n >= 2,
(2)... a(n+2) = 100^F(n-1,3)*a(n+1)^2 + a(n)
= 100^A006190(n-1)*a(n+1)^2 + a(n)
= 10^A052991(n-1)*a(n+1)^2 + a(n),
where F(n,3) is the n-th Fibonacci polynomial F(n,x) evaluated at x = 3.

cp's OEIS Frontend

A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Maxima

Formula