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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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

LINKS

Table of n, a(n) for n=0..65.

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

EXAMPLE

   [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

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}]]

PROG

(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)]

CROSSREFS

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

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

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

Sequence in context: A262082 A299045 A124530 * A070914 A305962 A144150

Adjacent sequences:  A243628 A243629 A243630 * A243632 A243633 A243634

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jun 08 2014

STATUS

approved

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Last modified March 25 22:28 EDT 2019. Contains 321477 sequences. (Running on oeis4.)