A243631 Square array of Narayana polynomials N_n evaluated at the integers, A(n,k) = N_n(k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 11, 14, 1, 1, 1, 5, 19, 45, 42, 1, 1, 1, 6, 29, 100, 197, 132, 1, 1, 1, 7, 41, 185, 562, 903, 429, 1, 1, 1, 8, 55, 306, 1257, 3304, 4279, 1430, 1, 1, 1, 9, 71, 469, 2426, 8925, 20071, 20793, 4862, 1

Offset: 0

Author

Peter Luschny, Jun 08 2014

Keywords

• nonn
• tabl

Comments

Mirror image of A008550. - Philippe Deléham, Sep 26 2014

Examples

			   [0]  [1]      [2]      [3]      [4]      [5]      [6]     [7]
[0] 1,   1,       1,       1,       1,       1,       1,       1
[1] 1,   1,       1,       1,       1,       1,       1,       1
[2] 1,   2,       3,       4,       5,       6,       7,       8 .. A000027
[3] 1,   5,      11,      19,      29,      41,      55,      71 .. A028387
[4] 1,  14,      45,     100,     185,     306,     469,     680 .. A090197
[5] 1,  42,     197,     562,    1257,    2426,    4237,    6882 .. A090198
[6] 1, 132,     903,    3304,    8925,   20076,   39907,   72528 .. A090199
[7] 1, 429,    4279,   20071,   65445,  171481,  387739,  788019 .. A090200
   A000108, A001003, A007564, A059231, A078009, A078018, A081178
First few rows of the antidiagonal triangle are:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2,  1;
  1, 1, 3,  5,  1;
  1, 1, 4, 11, 14,  1;
  1, 1, 5, 19, 45, 42, 1; - _G. C. Greubel_, Feb 16 2021
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Cf. A001263, A008550 (mirror), A204057 (another version), A242369 (main diagonal), A099169 (diagonal), A307883, A336727.

Rows[2-7]: A000027, A028387, A090197, A090198, A090199, A090200.

Columns[1-7]: A000108, A001003, A007564, A059231, A078009, A078018, A081178.

Cf. A132745.

Programs

Magma

A243631:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n,j)^2*k^j*(n-j)/(n*(j+1)): j in [0..n-1]]) >;
[A243631(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

Maple

# Computed with Narayana polynomials:
N := (n,k) -> binomial(n,k)^2*(n-k)/(n*(k+1));
A := (n,x) -> `if`(n=0, 1, add(N(n,k)*x^k, k=0..n-1));
seq(print(seq(A(n,k), k=0..7)), n=0..7);
# Computed by recurrence:
Prec := proc(n,N,k) option remember; local A,B,C,h;
if n = 0 then 1 elif n = 1 then 1+N+(1-N)*(1-2*k)
else h := 2*N-n; A := n*h*(1+N-n); C := n*(h+2)*(N-n);
B := (1+h-n)*(n*(1-2*k)*(1+h)+2*k*N*(1+N));
(B*Prec(n-1,N,k) - C*Prec(n-2,N,k))/A fi end:
T := (n, k) -> Prec(n,n,k)/(n+1);
seq(print(seq(T(n,k), k=0..7)), n=0..7);
# Array by o.g.f. of columns:
gf := n -> 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1):
for n from 0 to 11 do PolynomialTools:-CoefficientList(convert( series(gf(n), x, 12), polynom), x) od; # Peter Luschny, Nov 17 2014
# Row n by linear recurrence:
rec := n -> a(x) = add((-1)^(k+1)*binomial(n,k)*a(x-k), k=1..n):
ini := n -> seq(a(k) = A(n,k), k=0..n): # for A see above
row := n -> gfun:-rectoproc({rec(n),ini(n)},a(x),list):
for n from 1 to 7 do row(n)(8) od; # Peter Luschny, Nov 19 2014

Mathematica

MatrixForm[Table[JacobiP[n,1,-2*n-1,1-2*x]/(n+1), {n,0,7},{x,0,7}]]
Table[Hypergeometric2F1[1-k, -k, 2, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

Sage

def NarayanaPolynomial():
    R = PolynomialRing(ZZ, 'x')
    D = [1]
    h = 0
    b = True
    while True:
        if b :
            for k in range(h, 0, -1):
                D[k] += x*D[k-1]
            h += 1
            yield R(expand(D[0]))
            D.append(0)
        else :
            for k in range(0, h, 1):
                D[k] += D[k+1]
        b = not b
NP = NarayanaPolynomial()
for _ in range(8):
    p = next(NP)
    [p(k) for k in range(8)]

Sage

def A243631(n,k): return 1 if n==0 else sum( binomial(n,j)^2*k^j*(n-j)/(n*(j+1)) for j in [0..n-1])
flatten([[A243631(k,n-k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Feb 16 2021

Formula

T(n, k) = 2F1([1-n, -n], [2], k), 2F1 the hypergeometric function.

T(n, k) = P(n,1,-2*n-1,1-2*k)/(n+1), P the Jacobi polynomials.

T(n, k) = sum(j=0..n-1, binomial(n,j)^2*(n-j)/(n*(j+1))*k^j), for n>0.

For a recurrence see the second Maple program.

The o.g.f. of column n is gf(n) = 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1). - Peter Luschny, Nov 17 2014

T(n, k) ~ (sqrt(k)+1)^(2*n+1)/(2*sqrt(Pi)*k^(3/4)*n^(3/2)). - Peter Luschny, Nov 17 2014

The n-th row can for n>=1 be computed by a linear recurrence, a(x) = sum(k=1..n, (-1)^(k+1)*binomial(n,k)*a(x-k)) with initial values a(k) = p(n,k) for k=0..n and p(n,x) = sum(j=0..n-1, binomial(n-1,j)*binomial(n,j)*x^j/(j+1)) (implemented in the fourth Maple script). - Peter Luschny, Nov 19 2014

(n+1) * T(n,k) = (k+1) * (2*n-1) * T(n-1,k) - (k-1)^2 * (n-2) * T(n-2,k) for n>1. - Seiichi Manyama, Aug 08 2020

Sum_{k=0..n} T(k, n-k) = Sum_{k=0..n} 2F1([-k, 1-k], [2], n-k) = A132745(n). - G. C. Greubel, Feb 16 2021