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A243631
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Square array of Narayana polynomials N_n evaluated at the integers, A(n,k) = N_n(k), n>=0, k>=0, read by antidiagonals.
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9
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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
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OFFSET
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0,9
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COMMENTS
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LINKS
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FORMULA
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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
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EXAMPLE
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[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
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;
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MAPLE
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# 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):
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MATHEMATICA
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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 *)
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PROG
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(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])
(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]]) >;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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