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
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
Programs
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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
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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
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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
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