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A321620 The Riordan square of the Riordan numbers, triangle read by rows, T(n, k) for 0 <= k <= n. 38
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 6, 13, 10, 7, 4, 1, 1, 15, 29, 26, 16, 9, 5, 1, 1, 36, 73, 61, 42, 23, 11, 6, 1, 1, 91, 181, 157, 103, 61, 31, 13, 7, 1, 1, 232, 465, 398, 271, 156, 83, 40, 15, 8, 1, 1, 603, 1205, 1040, 702, 419, 221, 108, 50, 17, 9, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

If gf is a generating function of the sequence a then by the 'Riordan square of a' we understand the integer triangle given by the Riordan array (gf, gf). This mapping can be seen as an operator RS: Z[[x]] -> Mat[Z].

For instance A039599 is the Riordan square of the Catalan numbers, A172094 is the Riordan square of the little Schröder numbers and A063967 is the Riordan square of the Fibonacci numbers. The Riordan square of the simplest sequence of positive numbers (A000012) is the Pascal triangle.

The generating functions used are listed in the table below. They may differ slightly from those defined elsewhere in order to ensure that RS(a) is invertible (as a matrix). We understand this as a kind of normalization.

  ---------------------------------------------------------------------------

  Sequence a         | RS(a)   | gf(a)

  ---------------------------------------------------------------------------

  Catalan            | A039599 | (1 - sqrt(1 - 4*x))/(2*x).

  1, Riordan         | A321620 | 1 + 2*x/(1 + x + sqrt(1 - 2*x - 3*x^2)).

  Motzkin            | A321621 | (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2).

  Fine               | A321622 | 1 + (1 - sqrt(1 - 4*x))/(3 - sqrt(1 - 4*x)).

  large Schröder     | A321623 | (1 - x - sqrt(1 - 6*x + x^2))/(2*x).

  little Schröder    | A172094 | (1 + x - sqrt(1 - 6*x + x^2))/(4*x).

  Lucas              | A321624 | 1 + x*(1 + 2*x)/(1 - x - x^2).

  swinging factorial | A321625 | (1 + x/(1 - 4*x^2))/sqrt(1 - 4*x^2).

  ternary trees      ||A109956|| u = sqrt(x*3/4); sin(arcsin(3*u)/3)/u.

  central trinomial  | A116392 | 1/sqrt(1 - 2*x - 3*x^2)

  Bell               | A154380 | Sum_{k>=0} x^k/Product_{j=1..k}(1 - j*x).

  (2*n-1)!!          | A321627 | 1/(1-x/(1-2*x/(1-3*x/(1-4*x/(1-5*x/...

  powers of 2        | A038208 | 1/(1 - 2*x).

  the all 1's seq.   | A007318 | 1/(1 - x).

  Fibonacci          | A063967 | 1/(1 - x - x^2).

  tribonacci         | A187889 | 1/(1 - x - x^2 - x^3).

  tetranacci         | A353593 | 1/(1 - x - x^2 - x^3 - x^4).

  Jacobsthal         | A322942 | (2*x^2-1)/((x + 1)*(2*x - 1))

LINKS

Table of n, a(n) for n=0..77.

Peter Luschny, The Riordan Square Transform, 2018.

FORMULA

Given a generating function g and an positive integer N compute the Taylor expansion at the origin t(k) = [x^k] g(x) for k in [0...N-1] and set T(n, 0) = t(n) for n in [0...N-1]. Then compute T(m, k) = Sum_{j in [k-1...m-1]} T(j, k - 1) t(m - j) for k in [1...N-1] and for m in [k...N-1]. The resulting (0, 0)-based lower triangular array is the Riordan square generated by g.

EXAMPLE

The triangle starts:

   [ 0]   1

   [ 1]   1    1

   [ 2]   0    1    1

   [ 3]   1    1    1   1

   [ 4]   1    3    2   1   1

   [ 5]   3    5    5   3   1   1

   [ 6]   6   13   10   7   4   1   1

   [ 7]  15   29   26  16   9   5   1  1

   [ 8]  36   73   61  42  23  11   6  1  1

   [ 9]  91  181  157 103  61  31  13  7  1 1

   [10] 232  465  398 271 156  83  40 15  8 1 1

MAPLE

RiordanSquare := proc(d, n, exp:=false) local td, M, k, m, u, j;

    series(d, x, n+1); td := [seq(coeff(%, x, j), j = 0..n)];

    M := Matrix(n); for k from 1 to n do M[k, 1] := td[k] od;

    for k from 1 to n-1 do for m from k to n-1 do

       M[m+1, k+1] := add(M[j, k]*td[m-j+2], j = k..m) od od;

    if exp then u := 1;

       for k from 1 to n-1 do u := u * k;

          for m from 1 to k do j := `if`(m = 1, u, j/(m-1));

             M[k+1, m] := M[k+1, m] * j od od fi;

M end:

RiordanSquare(1 + 2*x/(1 + x + sqrt(1 - 2*x - 3*x^2)), 8);

MATHEMATICA

RiordanSquare[gf_, len_] := Module[{T}, T[n_, k_] := T[n, k] = If[k == 0, SeriesCoefficient[gf, {x, 0, n}], Sum[T[j, k-1] T[n-j, 0], {j, k-1, n-1}]]; Table[T[n, k], {n, 0, len-1}, {k, 0, n}]];

M = RiordanSquare[1 + 2x/(1 + x + Sqrt[1 - 2x - 3x^2]), 12];

M // Flatten (* Jean-François Alcover, Nov 24 2018 *)

PROG

(Sage) # uses[riordan_array from A256893]

def riordan_square(gf, len, exp=false):

    return riordan_array(gf, gf, len, exp)

riordan_square(1 + 2*x/(1 + x + sqrt(1 - 2*x - 3*x^2)), 10)

# Alternatively, given a list S:

def riordan_square_array(S):

    N = len(S)

    M = matrix(ZZ, N, N)

    for n in (0..N-1): M[n, 0] = S[n]

    for k in (1..N-1):

        for m in (k..N-1):

            M[m, k] = sum(M[j, k-1]*S[m-j] for j in (k-1..m-1))

    return M

riordan_square_array([1, 1, 0, 1, 1, 3, 6, 15, 36]) # Peter Luschny, Apr 03 2020

CROSSREFS

Row sums are A007971 and |A167022| shifted one place left.

Cf. A039599, A321621, A321622, A321624, A172094, A321623, A321624, A321625, A109956, A154380, A038208, A007318, A063967, A187889, A321629, A322942, A236376.

Sequence in context: A304972 A152176 A152175 * A353157 A134520 A188316

Adjacent sequences:  A321617 A321618 A321619 * A321621 A321622 A321623

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Nov 15 2018

STATUS

approved

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Last modified October 6 19:00 EDT 2022. Contains 357270 sequences. (Running on oeis4.)