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.

A078042 Expansion of (1-x)/(1+x-x^2+x^3).

Original entry on oeis.org

1, -2, 3, -6, 11, -20, 37, -68, 125, -230, 423, -778, 1431, -2632, 4841, -8904, 16377, -30122, 55403, -101902, 187427, -344732, 634061, -1166220, 2145013, -3945294, 7256527, -13346834, 24548655, -45152016, 83047505, -152748176, 280947697, -516743378, 950439251, -1748130326, 3215312955
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

Absolute values give coordination sequence for (3,infinity,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
a(n) is the upper left entry of the n-th power of the 3 X 3 matrix M = [-2, -2, 1; 1, 1, 0; 1, 0, 0]; a(n) = M^n [1, 1]. - Philippe Deléham, Apr 19 2023

Links

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

  • Magma
    [n le 3 select -n*(-1)^n else -Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015
  • Mathematica
    CoefficientList[Series[(1-x)/(1+x-x^2+x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,1,-1},{1,-2,3},40] (* Harvey P. Dale, Jun 01 2012 *)
  • PARI
    Vec((1-x)/(1+x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = -a(n-1) + a(n-2) - a(n-3) for n > 2; a(0)=1, a(1)=-2, a(2)=3. - Harvey P. Dale, Jun 01 2012
a(n) = (-1)^n * A001590(n+2).
a(n) = Sum_{k=0..n} A188316(n,k)*(-2)^k. - Philippe Deléham, Apr 19 2023

A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 9, 16, 13, 5, 1, 1, 12, 30, 32, 19, 6, 1, 1, 16, 50, 71, 55, 26, 7, 1, 1, 20, 80, 140, 140, 86, 34, 8, 1, 1, 25, 120, 259, 316, 246, 126, 43, 9, 1, 1, 30, 175, 448, 660, 622, 399, 176, 53, 10, 1
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is (1/(1-x*z/((1-z^2)*(1-z))))/(1-z).
Row sums give A052534. Column sequences (without leading zeros) give A000012 (powers of 1), A002620(n+1), A002624, A060099-A060101 for m=0..5.
The bisections of the column sequences give triangles A060102 and A060556.
Riordan array (1/(1-x), x/((1-x)*(1-x^2))). - Paul Barry, Mar 28 2011

Examples

			p(3,x) = 1 + 4*x + 3*x^2 + x^3.
Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  6,  8,  4,  1;
  1,  9, 16, 13,  5,  1;
  1, 12, 30, 32, 19,  6,  1;
  1, 16, 50, 71, 55, 26,  7,  1;
  ...

Links

Crossrefs

Cf. A052534, A000012, A002620(n+1), A002624, A060099, A060100, A060101.
Cf. A060102, A060556, A188316.

Programs

  • Maple
    A060098 := proc(n,k) add( binomial(n-2*i,n-2*i-k)*binomial(k+i-1,i), i=0..floor(n/2)) ; end proc:
seq(seq(A060098(n,k), k=0..n), n=0..12); # R. J. Mathar, Mar 29 2011
# Recurrence after Philippe Deléham:
T := proc(n, k) option remember;
if k < 0 or k > n then 0 elif k = 0 or n = k then 1 else
T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, May 07 2023
  • Mathematica
    t[n_, k_] := Sum[ Binomial[n-2*j, n-2*j-k]*Binomial[k+j-1, j], {j, 0, n/2}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

Formula

G.f. for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/(1-x) = x^m/((1+x)^m*(1-x)^(2*m+1)).
Number triangle T(n,k) = Sum_{i=0..floor(n/2)} C(n-2*i,n-2*i-k)*C(k+i-1,i). - Paul Barry, Mar 28 2011
From Philippe Deléham, Apr 20 2023: (Start)
T(n, k) = 0 if k < 0 or if k > n; T(n, k) = 1 if k = 0 or k = n; otherwise:
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k).
T(n, k) = A188316(n, k) + A188316(n-1, k). (End)

A188312 Expansion of (1/(1-x^2))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 4, 12, 45, 174, 709, 2978, 12825, 56303, 251060, 1133943, 5176926, 23851690, 110759081, 517853840, 2435786531, 11517940357, 54722081630, 261089977806, 1250479470053, 6009884614944, 28975052979797, 140098515402139, 679189779433800, 3300702453217325, 16076773046682690
Offset: 0

Views

Author

Paul Barry, Mar 28 2011

Keywords

Comments

Hankel transform is the (25,-29) Somos-4 sequence A188313. Image of Catalan numbers by A188316.

Links

Crossrefs

Cf. A188314.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x^2 -Sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)))); // G. C. Greubel, Aug 14 2018
  • Maple
    a := n -> add((-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4), i=0..n);
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 03 2018
  • Mathematica
    CoefficientList[Series[(1-x^2 -Sqrt[1-4*x-6*x^2+x^4])/(2*x*(1+x)), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
  • Maxima
    a(n):=sum(sum((-1)^(n-k-i)*binomial(k+i-1, k-1)*binomial(2*k+i-2, k+i-1)* binomial(n-i-1, n-k-i)/k,k,1,n-i),i,0,n); /* Vladimir Kruchinin, May 03 2018 */
  • PARI
    x='x+O('x^50); Vec((1-x^2 -sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x))) \\ G. C. Greubel, Aug 14 2018

Formula

G.f.: (1-x^2 - sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)).
G.f.: u(x)=1/(1-x^2-x/(1-x-x*u(x))).
G.f.: 1/(1-x^2-x/(1-x-x/(1-x^2-x/(1-x-x/(1-...))))) (continued fraction).
Conjecture: (n+1)*a(n) +(3-4*n)*a(n-1) + (7-6*n)*a(n-2) -a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2011
a(n) = a(n-1) + (-1)^n + Sum_{i=0..n-1} a(i)*a(n-1-i). - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n} Sum_{k=1..n-i} (-1)^(n-k-i)*C(k+i-1,k-1)*C(2*k+i-2,k+i-1)*C(n-i-1,n-k-i)/k. - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n} (-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4). - Peter Luschny, May 03 2018

A078058 Expansion of (1-x)/(1+2*x-x^2+x^3).

Original entry on oeis.org

1, -3, 7, -18, 46, -117, 298, -759, 1933, -4923, 12538, -31932, 81325, -207120, 527497, -1343439, 3421495, -8713926, 22192786, -56520993, 143948698, -366611175, 933692041, -2377943955, 6056191126, -15424018248, 39282171577, -100044552528, 254795294881, -648917313867
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

a(n) is the upper left entry of the n-th power of the 3 X 3 matrix M = [-3, -3, 1; 1, 1, 0; 1, 0, 0]; a(n) = M^n [1, 1]. - Philippe Deléham, Apr 19 2023

Links

Programs

  • Mathematica
    CoefficientList[Series[(1-x)/(1+2x-x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{-2,1,-1},{1,-3,7},31] (* Harvey P. Dale, Oct 22 2011 *)
  • PARI
    Vec((1-x)/(1+2*x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

a(n) = -2*a(n-1) + a(n-2) - a(n-3) for n > 2; a(0) = 1, a(1) = -3, a(2) = 7. - Harvey P. Dale, Oct 22 2011
a(n) = Sum_{k = 0..n} A188316(n, k)*(-3)^k. - Philippe Deléham, Apr 19 2023
a(n) = A077986(n)-A077986(n-1) . - R. J. Mathar, Mar 19 2025

A188317 Riordan array, inverse of (1/(1-x^2), x/((1-x)*(1-x^2))).

Original entry on oeis.org

1, 0, 1, -1, -1, 1, 2, -1, -2, 1, -1, 6, 0, -3, 1, -6, -10, 10, 2, -4, 1, 20, -4, -26, 13, 5, -5, 1, -22, 63, 14, -49, 14, 9, -6, 1, -49, -148, 112, 58, -78, 12, 14, -7, 1, 260, 45, -396, 144, 138, -111, 6, 20, -8, 1, -441, 755, 435, -789, 123, 263, -145, -5, 27, -9, 1
Offset: 0

Views

Author

Paul Barry, Mar 28 2011

Keywords

Comments

Inverse of number triangle A188316.

Examples

			Triangle begins
1,
0, 1,
-1, -1, 1,
2, -1, -2, 1,
-1, 6, 0, -3, 1,
-6, -10, 10, 2, -4, 1,
20, -4, -26, 13, 5, -5, 1,
-22, 63, 14, -49, 14, 9, -6, 1,
-49, -148, 112, 58, -78, 12, 14, -7, 1,
260, 45, -396, 144, 138, -111, 6, 20, -8, 1
Production matrix begins
0, 1,
-1, -1, 1,
1, -1, -1, 1,
0, 1, -1, -1, 1,
0, 0, 1, -1, -1, 1,
0, 0, 0, 1, -1, -1, 1,
0, 0, 0, 0, 1, -1, -1, 1,
0, 0, 0, 0, 0, 1, -1, -1, 1,
0, 0, 0, 0, 0, 0, 1, -1, -1, 1
Showing 1-5 of 5 results.

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

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