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.

A115148 Ninth convolution of A115140.

Original entry on oeis.org

1, -9, 27, -30, 9, 0, 0, 0, 0, -1, -9, -54, -273, -1260, -5508, -23256, -95931, -389367, -1562275, -6216210, -24582285, -96768360, -379629720, -1485507600, -5801732460, -22626756594, -88152205554, -343176898988, -1335293573130, -5193831553416
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A115139 - A115147, A115149.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
  • Mathematica
    CoefficientList[Series[(1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3 +x^4)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2 -10*x^3+x^4)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
  • Sage
    ((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4) *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)^9 = P(10, x) - x*P(9, 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(10, x)=1-8*x+21*x^2-20*x^3+5*x^4 and P(9, x)=1-7*x+15*x^2-10*x^3+x^4.
a(n) = -C9(n-9), n>=9, with C9(n) = A001392(n+4) (eighth convolution of Catalan numbers). a(0)=1, a(1)=-9, a(2)=27, a(3)=-30, a(4)=9, a(5)=a(6)=a(7)=a(8)=0. [1, -9, 27, -30, 9] is row n=9 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

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

A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 0, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796
Offset: 0

Views

Author

Peter Luschny, Jun 25 2022

Keywords

Comments

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 1. (See the Python program for a reference implementation.)
This definition also includes the classical Fuss-Catalan numbers, since T(n, n) = A000108(n), or row 2 in A355262. For m = 2 see A355172 and for m = 3 A355174. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

Examples

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 2]
  [3] [0, 1, 3,  5]
  [4] [0, 1, 4,  9,  14]
  [5] [0, 1, 5, 14,  28,  42]
  [6] [0, 1, 6, 20,  48,  90,  132]
  [7] [0, 1, 7, 27,  75, 165,  297, 429]
  [8] [0, 1, 8, 35, 110, 275,  572, 1001, 1430]
  [9] [0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1, 2,  5,  14,   42,  132,   429,  1430,   4862,   16796, ...  A000108
  [1] 0, 1, 3,  9,  28,   90,  297,  1001,  3432,  11934,   41990, ...  A000245
  [2] 0, 1, 4, 14,  48,  165,  572,  2002,  7072,  25194,   90440, ...  A099376
  [3] 0, 1, 5, 20,  75,  275, 1001,  3640, 13260,  48450,  177650, ...  A115144
  [4] 0, 1, 6, 27, 110,  429, 1638,  6188, 23256,  87210,  326876, ...  A115145
  [5] 0, 1, 7, 35, 154,  637, 2548,  9996, 38760, 149226,  572033, ...  A000588
  [6] 0, 1, 8, 44, 208,  910, 3808, 15504, 62016, 245157,  961400, ...  A115147
  [7] 0, 1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, ...  A115148

Crossrefs

A000108 (main diagonal), A000245 (subdiagonal), A002057 (diagonal 2), A000344 (diagonal 3), A000027 (column 2), A000096 (column 3), A071724 (row sums), A000958 (alternating row sums), A262394 (main diagonal of array).
Variants: A009766 (main variant), A030237, A130020.
Cf. A123110 (triangle of order 0), A355172 (triangle of order 2), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
Cf. A115144, A115145, A000588, A115147, A115148.

Programs

  • Python
    from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(row))
for n in range(11): print(Trow(n))

Formula

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 1).
T(n, k) = (n - k + 2)*(n + k - 1)!/((n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 2*x^2)/(1 - x)^(n + 2)).

A368380 Arises from enumeration of a certain class of partial zig-zag knight's paths on the square grid.

Original entry on oeis.org

0, 0, 0, 1, 0, 5, 1, 20, 8, 75, 44, 275, 208, 1001, 910, 3640, 3808, 13260, 15504, 48450, 62016, 177650, 245157, 653752, 961400, 2414425, 3749460, 8947575, 14567280, 33266625, 56448210, 124062000, 218349120, 463991880, 843621600, 1739969550, 3257112960
Offset: 0

Views

Author

N. J. A. Sloane, Feb 18 2024

Keywords

Comments

It would be nice to have a more precise definition.

Links

Crossrefs

Cf. A368378, A368379.
The two bisections are A115144 (shifted, negated) and A115147 (shifted, negated).

Formula

G.f.: (1/x + 1 + 2*R(x) + R(x)^2) * R(x)^3 + R(x)^2 / x = F(x) * R(x), where R(x) = (1 - sqrt(1-4*x^2)) / (2*x^2) - 1 and F(x) is the g.f. of A368379. - Andrei Zabolotskii, Jul 25 2025

Extensions

Terms a(13) and beyond from Andrei Zabolotskii, Jul 25 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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