A154955 -id:A154955 - cp's OEIS frontend

A143259 a(n) = 1 if n is a nonzero square, -1 if n is twice a nonzero square, 0 otherwise.

1, -1, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Michael Somos, Aug 02 2008

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - q^2 + q^4 - q^8 + q^9 + q^16 - q^18 + q^25 - q^32 + q^36 + q^49 - q^50 + ...

Antti Karttunen, Table of n, a(n) for n = 1..65537
S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 1.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008836, A010052, A053866, A111374, A154955.

Basis( ModularForms( Gamma1(8), 1/2), 100) [2] ; /* Michael Somos, Jun 10 2014 */

f[n_]:=Which[IntegerQ[Sqrt[n/2]],-1,IntegerQ[Sqrt[n]],1,True,0]; Array[f,110] (* Harvey P. Dale, Jul 07 2011 *)
a[ n_] := Boole[ IntegerQ[ Sqrt[n]]] - Boole[ IntegerQ[ Sqrt[2 n]]]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2])/2, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)
Table[LiouvilleLambda[n]*Mod[DivisorSigma[1, n], 2], {n, 100}] (* Jon Maiga, Jan 11 2019 *)

PARI
```
{a(n) = issquare(n) - issquare(2*n)};
```

{a(n) = if( n<1, 0, n--; polcoeff( prod(k=1, n, (1 - x^k)^([1, 1, 0, -1, -1, -1, 0, 1][k%8 + 1]), 1 + x * O(x^n)), n))};

Expansion of (phi(q) - phi(q^2)) / 2 = q * psi(q^4) * f(-q, -q^7) / f(-q^3, -q^5) in powers of q where phi(), psi() and f() are Ramanujan theta functions.
Expansion of q * f(-q, -q^7)^2 / psi(-q) in powers of q where psi(), f() are Ramanujan theta functions. - Michael Somos, Jan 01 2015
Euler transform of period 8 sequence [ -1, 0, 1, 1, 1, 0, -1, -1, ...].
a(2*n) = -a(n) for all n in Z.
a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 1 (mod 2).
Dirichlet g.f.: zeta(2*s) * (1 - 2^-s); Dirichlet convolution of A010052 and A154955.
G.f. A(x) satisfies: A(x) / A(x^2) = -1 + A111374(x).
G.f. A(x) satisfies: A(x^2) = - (A(x) + A(-x)) / 2.
G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u + v)^2 - v * (v + w) * (v + 2*w).
G.f.: (theta_3(q) - theta_3(q^2)) / 2 = Sum_{k>0} x^(k^2) - x^(2k^2).
|a(n)| = A053866(n).
a(n) = A008836(n)*A053866(n). - Jon Maiga, Jan 11 2019
Sum_{k=1..n} a(k) ~ (1 - 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020

A352687 Triangle read by rows, a Narayana related triangle whose rows are refinements of twice the Catalan numbers (for n >= 2).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 4, 1, 0, 1, 7, 12, 7, 1, 0, 1, 11, 30, 30, 11, 1, 0, 1, 16, 65, 100, 65, 16, 1, 0, 1, 22, 126, 280, 280, 126, 22, 1, 0, 1, 29, 224, 686, 980, 686, 224, 29, 1, 0, 1, 37, 372, 1512, 2940, 2940, 1512, 372, 37, 1

Offset: 0

Peter Luschny, Apr 26 2022

This is the second triangle in a sequence of Narayana triangles. The first is A090181, whose n-th row is a refinement of Catalan(n), whereas here the n-th row of T is a refinement of 2*Catalan(n-1). We can show that T(n, k) <= A090181(n, k) for all n, k. The third triangle in this sequence is A353279, where also a recurrence for the general case is given.
Here we give a recurrence for the row polynomials, which correspond to the recurrence of the classical Narayana polynomials combinatorially proved by Sulanke (see link).
The polynomials have only real zeros and form a Sturm sequence. This follows from the recurrence along the lines given in the Chen et al. paper.
Some interesting sequences turn out to be the evaluation of the polynomial sequence at a fixed point (see the cross-references), for example the reversion of the Jacobsthal numbers A001045 essentially is -(-2)^n*P(n, -1/2).
The polynomials can also be represented as the difference between generalized Narayana polynomials, see the formula section.

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 1,  2,   1;
[4] 0, 1,  4,   4,   1;
[5] 0, 1,  7,  12,   7,   1;
[6] 0, 1, 11,  30,  30,  11,   1;
[7] 0, 1, 16,  65, 100,  65,  16,   1;
[8] 0, 1, 22, 126, 280, 280, 126,  22,  1;
[9] 0, 1, 29, 224, 686, 980, 686, 224, 29, 1;

Xi Chen, Arthur Li Bo Yang, James Jing Yu Zhao, Recurrences for Callan's Generalization of Narayana Polynomials, J. Syst. Sci. Complex (2021).
Peter Luschny, Illustration of the polynomials.
Robert A. Sulanke, The Narayana distribution, J. Statist. Plann. Inference, 2002, 101: 311-326, formula 2.

Cf. A090181 and A001263 (Narayana), A353279 (case 3), A000108 (Catalan), A145596, A172392 (central terms), A000124 (subdiagonal, column 2), A115143.
Essentially twice the Catalan numbers: A284016 (also A068875, A002420).
Values of the polynomial sequence: A068875 (row sums): P(1), A154955: P(-1), A238113: P(2)/2, A125695 (also A152681): P(-2), A054872: P(3)/2, P(3)/6 probable A234939, A336729: P(-3)/6, A082298: P(4)/5, A238113: 2^n*P(1/2), A154825 and A091593: 2^n*P(-1/2).

T := (n, k) -> if n = k then 1 elif k = 0 then 0 else
binomial(n, k)^2*(k*(2*k^2 + (n + 1)*(n - 2*k))) / (n^2*(n - 1)*(n - k + 1)) fi:
seq(seq(T(n, k), k = 0..n), n = 0..10);
# Alternative:
gf := 1 - x + (1 + y)*(1 - x*(y - 1) - sqrt((x*y + x - 1)^2 - 4*x^2*y))/2:
serx := expand(series(gf, x, 16)): coeffy := n -> coeff(serx, x, n):
seq(seq(coeff(coeffy(n), y, k), k = 0..n), n = 0..10);
# Using polynomial recurrence:
P := proc(n, x) option remember; if n < 3 then [1, x, x + x^2] [n + 1] else
((2*n - 3)*(x + 1)*P(n - 1, x) - (n - 3)*(x - 1)^2*P(n - 2, x)) / n fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k = 0..n): seq(Trow(n), n = 0..10);
# Represented by generalized Narayana polynomials:
N := (n, k, x) -> add(((k+1)/(n-k))*binomial(n-k,j-1)*binomial(n-k,j+k)*x^(j+k), j=0..n-2*k): seq(print(ifelse(n=0, 1, expand(N(n,0,x) - N(n,1,x)))), n=0..7);

H[0, ] := 1; H[1, x] := x;
H[n_, x_] := x*(x + 1)*Hypergeometric2F1[1 - n, 2 - n, 2, x];
Hrow[n_] := CoefficientList[H[n, x], x]; Table[Hrow[n], {n, 0, 9}] // TableForm

from math import comb as binomial
def T(n, k):
    if k == n: return 1
    if k == 0: return 0
    return ((binomial(n, k)**2 * (k * (2 * k**2 + (n + 1) * (n - 2 * k))))
           // (n**2 * (n - 1) * (n - k + 1)))
def Trow(n): return [T(n, k) for k in range(n + 1)]
for n in range(10): print(Trow(n))

# The recursion with cache is (much) faster:
from functools import cache
@cache
def T_row(n):
    if n < 3: return ([1], [0, 1], [0, 1, 1])[n]
    A = T_row(n - 2) + [0, 0]
    B = T_row(n - 1) + [1]
    for k in range(n - 1, 1, -1):
        B[k] = (((B[k] + B[k - 1]) * (2 * n - 3)
               - (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 3)) // n)
    return B
for n in range(10): print(T_row(n))

Explicit formula (additive form):
T(n, n) = 1, T(n > 0, 0) = 0 and otherwise T(n, k) = binomial(n, k)*binomial(n - 1, k - 1)/(n - k + 1) - 2*binomial(n - 1, k)*binomial(n - 1, k - 2)/(n - 1).
Multiplicative formula with the same boundary conditions:
T(n, k) = binomial(n, k)^2*(k*(2*k^2 + (n + 1)*(n - 2*k)))/(n^2*(n-1)*(n- k + 1)).
Bivariate generating function:
T(n, k) = [x^n] [y^k](1 - x + (1+y)*(1-x*(y-1) - sqrt((x*y+x-1)^2 - 4*x^2*y))/2).
Recursion based on polynomials:
T(n, k) = [x^k] (((2*n - 3)*(x + 1)*P(n - 1, x) - (n - 3)*(x - 1)^2*P(n - 2, x)) / n) with P(0, x) = 1, P(1, x) = x, and P(2, x) = x + x^2.
Recursion based on rows (see the second Python program):
T(n, k) = (((B(k) + B(k-1)) * (2*n - 3) - (A(k) - 2*A(k-1) + A(k-2))*(n-3))/n), where A(k) = T(n-2, k) and B(k) = T(n-1, k), for n >= 3.
Hypergeometric representation:
T(n, k) = [x^k] x*(x + 1)*hypergeom([1 - n, 2 - n], [2], x) for n >= 2.
Row sums:
Sum_{k=0..n} T(n, k) = (2/n)*binomial(2*(n - 1), n - 1) = A068875(n-1) for n >= 2.
A generalization of the Narayana polynomials is given by
N{n, k}(x) = Sum_{j=0..n-2*k}(((k + 1)/(n - k)) * binomial(n - k, j - 1) * binomial(n - k, j + k) * x^(j + k)).
N{n, 0}(x) are the classical Narayana polynomials A001263 and N{n, 1}(x) is a shifted version of A145596 based in (3, 2). Our polynomials are the difference P(n, x) = N{n, 0}(x) - N{n, 1}(x) for n >= 1.
Let RS(T, n) denote the row sum of the n-th row of T, then RS(T, n) - RS(A090181, n) = -4*binomial(2*n - 3, n - 3)/(n + 1) = A115143(n + 1) for n >= 3.

A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

Original entry on oeis.org

1, 2, -1, 3, -3, 0, 4, -6, 2, 0, 5, -10, 8, -3, 0, 6, -15, 20, -18, 6, 0, 7, -21, 40, -60, 48, -11, 0, 8, -28, 70, -150, 204, -124, 18, 0, 9, -36, 112, -315, 624, -690, 312, -30, 0, 10, -45, 168, -588, 1554, -2620, 2340, -810, 56, 0, 11, -55, 240, -1008, 3360, -7805, 11160, -8160, 2184, -105, 0

Offset: 1

Seiichi Manyama, Apr 12 2025

			Square array begins:
   1,   2,    3,    4,     5,     6,      7, ...
  -1,  -3,   -6,  -10,   -15,   -21,    -28, ...
   0,   2,    8,   20,    40,    70,    112, ...
   0,  -3,  -18,  -60,  -150,  -315,   -588, ...
   0,   6,   48,  204,   624,  1554,   3360, ...
   0, -11, -124, -690, -2620, -7805, -19656, ...
   0,  18,  312, 2340, 11160, 39990, 117648, ...

Columns k=1..5 give A154955, A038063, A038064, A038065, A038066.
Main diagonal gives A383012.
Cf. A008683, A074650.

a(n, k) = -sumdiv(n, d, moebius(n/d)*(-k)^d)/n;

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + k*x^j) / j.

Product_{n>=1} 1/(1 - x^n)^A(n,k) = 1 + k*x.

A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 5, 1, 0, 1, 20, 21, 7, 1, 0, 1, 48, 81, 40, 9, 1, 0, 1, 112, 297, 208, 65, 11, 1, 0, 1, 256, 1053, 1024, 425, 96, 13, 1, 0, 1, 576, 3645, 4864, 2625, 756, 133, 15, 1, 0, 1, 1280, 12393, 22528, 15625, 5616, 1225, 176, 17, 1

Offset: 0

Stefano Spezia, Jan 31 2025

			The array begins as:
   1, 1,   1,    1,     1,     1, ...
  -1, 1,   3,    5,     7,     9, ...
   0, 1,   8,   21,    40,    65, ...
   0, 1,  20,   81,   208,   425, ...
   0, 1,  48,  297,  1024,  2625, ...
   0, 1, 112, 1053,  4864, 15625, ...
   0, 1, 256, 3645, 22528, 90625, ...
   ...

Cf. A000012 (k=1 or n=0), A000567 (n=2), A001792 (k=2), A007778, A060747 (n=1), A081038 (k=3), A081039 (k=4), A081040 (k=5), A081041 (k=6), A081042 (k=7), A081043 (k=8), A081044 (k=9), A081045 (k=10), A103532, A154955, A380748 (antidiagonal sums).

A[0,0]:=1; A[1,0]:=-1; A[n_,k_]:=((k-1)*n+k)k^(n-1); Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1-x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+(k-1)*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = ((k - 1)*n + k)*k^(n-1) with A(0,0) = 1.
A(n,k) = n! * [x^n] exp(k*x)*(1 + (k - 1)*x).
A(n,0) = A154955(n+1).
A(3,n) = A103532(n-1) for n > 0.
A(n,n) = A007778(n) for n > 0.

cp's OEIS Frontend

A143259 a(n) = 1 if n is a nonzero square, -1 if n is twice a nonzero square, 0 otherwise.

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A352687 Triangle read by rows, a Narayana related triangle whose rows are refinements of twice the Catalan numbers (for n >= 2).

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A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

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A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

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