cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A115142 Third convolution of A115140.

Original entry on oeis.org

1, -3, 0, -1, -3, -9, -28, -90, -297, -1001, -3432, -11934, -41990, -149226, -534888, -1931540, -7020405, -25662825, -94287120, -347993910, -1289624490, -4796857230, -17902146600, -67016296620, -251577050010, -946844533674, -3572042254128, -13505406670700
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A000108, A000245, A071724.
Cf. A115139, A115140, A115141, A115143 - A115149.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2*(1-2*x)-(1-x)*(1-Sqrt(1-4*x)))/2 )); // G. C. Greubel, Feb 12 2019
  • Mathematica
    CoefficientList[Series[(2*(1-2*x)-(1-x)*(1-Sqrt[1-4*x]))/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((2*(1-2*x)-(1-x)*(1-sqrt(1-4*x)))/2) \\ G. C. Greubel, Feb 12 2019
  • Sage
    ((2*(1-2*x)-(1-x)*(1-sqrt(1-4*x)))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Formula

O.g.f.: 1/c(x)^3 = P(4, x) - x*P(3, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(4, x)=1-2*x and P(3, x)=1-x.
a(n) = -C3(n-3), n >= 3, with C3(n):= A000245(n+1) (third convolution of Catalan numbers). a(0)=1, a(1)=-3, a(2)=0. [1, -3] is the row n=3 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-3)*a(n) -2*(n-2)*(2*n-5)*a(n-1)=0. - R. J. Mathar, Sep 23 2021
O.g.f.: (1/8)*(1 + sqrt(1 - 4*x))^3 = ( hypergeom([-1/4, -3/4], [-1/2], 4*x) )^2. - Peter Bala, Mar 04 2022

A115144 Fifth convolution of A115140.

Original entry on oeis.org

1, -5, 5, 0, 0, -1, -5, -20, -75, -275, -1001, -3640, -13260, -48450, -177650, -653752, -2414425, -8947575, -33266625, -124062000, -463991880, -1739969550, -6541168950, -24647883000, -93078189750, -352207870014, -1335293573130, -5071418015120, -19293438101000
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A000344, A115139 - A115143, A115145 - A115149.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
  • Mathematica
    CoefficientList[Series[(1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
  • Sage
    ((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Formula

O.g.f.: 1/c(x)^5 = P(6, x) - x*P(5, x)*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(6, x)=1-4*x+3*x^2 and P(5, x)=1-3*x+x^2.
a(n) = -C5(n-5), n>=5, with C5(n) = A000344(n+2) (fifth convolution of Catalan numbers). a(0)=1, a(1)=-5, a(2)=5, a(3)=0=a(4). [1, -5, 5] is row n=5 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-5)*a(n) -2*(n-3)*(2*n-7)*a(n-1)=0. - R. J. Mathar, Sep 23 2021
From Peter Bala, Mar 05 2023: (Start)
a(n) = binomial(2*n - 6, n) - binomial(2*n - 6, n + 1).
a(n) = = -5/(n - 5)*binomial(2*n - 6, n) for n != 5.
a(n) = -A000344(n-3) for n >= 5. (End)

A115145 Sixth convolution of A115140.

Original entry on oeis.org

1, -6, 9, -2, 0, 0, -1, -6, -27, -110, -429, -1638, -6188, -23256, -87210, -326876, -1225785, -4601610, -17298645, -65132550, -245642760, -927983760, -3511574910, -13309856820, -50528160150, -192113383644, -731508653106, -2789279908316, -10649977831752, -40715807302800
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A115139, A115140, A115141, A115142, A115143.
Cf. A115144, A115146, A115147, A115148, A115149.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
  • Mathematica
    CoefficientList[Series[(1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
  • Sage
    ((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Formula

O.g.f.: 1/c(x)^6 = P(7, x) - x*P(6, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(7, x)=1-5*x+6*x^2-x^3 and P(6, x) = 1-4*x+3*x^2.
a(n) = -C6(n-6), n>=6, with C6(n) = A003517(n+2) (sixth convolution of Catalan numbers). a(0)=1, a(1)=-6, a(2)=9, a(3)=-2, a(4)=0=a(5). [1, -6, 9, -2] is row n=6 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-6)*a(n) -2*(2*n-7)*(n-4)*a(n-1)=0. - R. J. Mathar, Sep 23 2021

A115146 Seventh convolution of A115140.

Original entry on oeis.org

1, -7, 14, -7, 0, 0, 0, -1, -7, -35, -154, -637, -2548, -9996, -38760, -149226, -572033, -2187185, -8351070, -31865925, -121580760, -463991880, -1771605360, -6768687870, -25880277150, -99035193894, -379300783092, -1453986335186, -5578559816632, -21422369201800
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A115139, A115140, A115141, A115142, A115143.
Cf. A115144, A115145, A115147, A115148, A115149.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
  • Mathematica
    CoefficientList[Series[(1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
  • Sage
    ((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*sqrt(1-4*x))/2 ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Formula

O.g.f.: 1/c(x)^7 = P(8, x) - x*P(7, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(8, x)=1-6*x+10*x^2-4*x^3 and P(7, x)=1-5*x+6*x^2-x^3.
a(n) = -C7(n-7), n>=7, with C7(n):=A000588(n+3) (seventh convolution of Catalan numbers). a(0)=1, a(1)=-7, a(2)=14, a(3)=-7, a(4)=a(5)=a(6)=0. [1, -7, 14, -7] is row n=7 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence n*(n-7)*a(n) -2*(n-4)*(2*n-9)*a(n-1)=0. - R. J. Mathar, Sep 15 2024

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

Views

Author

Peter Luschny, Apr 26 2022

Keywords

Comments

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.

Examples

			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;

Links

Crossrefs

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).

Programs

  • Maple
    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);
  • Mathematica
    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
  • Python
    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))
  • Python
    # 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))

Formula

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.
Showing 1-5 of 5 results.

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