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 1-4 of 4 results.

A249059 Row sums of the triangular array at A249057.

Original entry on oeis.org

1, 5, 10, 40, 110, 430, 1420, 5720, 21340, 89980, 367400, 1627120, 7138120, 33172040, 154520080, 751616800, 3687498320, 18719834320, 96157299040, 507993654080, 2719611532000, 14911459229920, 82901747529920, 470599687507840, 2708946870815680
Offset: 0

Views

Author

Clark Kimberling, Oct 20 2014

Keywords

Examples

			First 3 rows from A249057:
1
4    1
5    4    1,
so that a(0) = 1, a(1) = 5, a(2) = 10.

Links

Crossrefs

Cf. A249057, A249060.

Programs

  • Mathematica
    z = 30; p[x_, n_] := x + (n + 2)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}];
u = Numerator[t];
v1 = Flatten[CoefficientList[u, x]]; (* A249057  *)
v2 = u /. x -> 1  (* A249059 *)
v3 = u /. x -> 0  (* A249060 *)

A249060 Column 1 of the triangular array at A249057.

Original entry on oeis.org

1, 4, 5, 24, 35, 192, 315, 1920, 3465, 23040, 45045, 322560, 675675, 5160960, 11486475, 92897280, 218243025, 1857945600, 4583103525, 40874803200, 105411381075, 980995276800, 2635284526875, 25505877196800, 71152682225625, 714164561510400, 2063427784543125
Offset: 0

Views

Author

Clark Kimberling, Oct 20 2014

Keywords

Examples

			First 3 rows from A249057:
1
4    1
5    4    1,
so that a(0) = 1, a(1) = 4, a(2) = 5.

Links

Crossrefs

Cf. A178647, A249057, A309491.

Programs

  • Mathematica
    z = 30; p[x_, n_] := x + (n + 2)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}];
u = Numerator[t]; v1 = Flatten[CoefficientList[u, x]]; (* A249057 *)
v2 = u /. x -> 1  (* A249059 *)
v3 = u /. x -> 0  (* A249060 *)
  • PARI
    f(n) = if (n, x + (n + 3)/f(n-1), 1);
a(n) = polcoef(numerator(f(n)), 0); \\ Michel Marcus, Nov 25 2022

Formula

From Derek Orr, Oct 21 2014: (Start)
a(2*n) = (2*n+3)*(2*n+1)!!/3, for n > 0.
a(2*n+1) = (n+2)!*2^(n+1), for n > 0.
For n > 2, if n is even, a(n)/[(n+1)*(n-1)*(n-3)*...*7*5] = n + 3 and if n is odd, a(n)/[(n+1)*(n-1)*(n-3)*...*6*4] = n + 3. (End)
a(n) = gcd_2((n+3)!,(n+3)!!), where gcd_2(b,c) denotes the second-largest common divisor of non-coprime integers b and c, as defined in A309491. - Lechoslaw Ratajczak, Apr 15 2021
D-finite with recurrence: a(n) - (3+n)*a(n-2) = 0. - Georg Fischer, Nov 25 2022
Sum_{n>=0} 1/a(n) = 3*sqrt(e*Pi/2)*erf(1/sqrt(2)) + 2*sqrt(e) - 6, where erf is the error function. - Amiram Eldar, Dec 10 2022

A230698 Triangle read by rows: T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 8, 7, 9, 1, 1, 15, 33, 12, 14, 1, 1, 48, 57, 87, 18, 20, 1, 1, 105, 279, 141, 185, 25, 27, 1, 1, 384, 561, 975, 285, 345, 33, 35, 1, 1, 945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1, 3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1
Offset: 0

Views

Author

Philippe Deléham, Oct 28 2013

Keywords

Comments

Triangle A180048 mixed with triangle A180049.
Let p(n,x) be the polynomial whose coefficients are given by row n; e.g., p(2,x) = 2 + x + x^2; then p(n,x) is the numerator of the rational function given by f(n,x) = x + (n - 1)/f(n-1,x), where f(x,0) = 1. (Sum of numbers in row n) = A000885(n) for n >= 1. (Column 1) = A006882 (n-th term = n!! for n >= 0) - Clark Kimberling, Oct 19 2014

Examples

			Triangle begins (0<=k<=n):
1
1, 1
2, 1, 1
3, 5, 1, 1
8, 7, 9, 1, 1
15, 33, 12, 14, 1, 1
48, 57, 87, 18, 20, 1, 1
105, 279, 141, 185, 25, 27, 1, 1
384, 561, 975, 285, 345, 33, 35, 1, 1
945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1
3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1
10395, 35685, 26685, 41685, 10290, 12558, 1302, 1422, 63, 65, 1, 1

Links

Crossrefs

Cf. A180048, A180049.

Programs

  • Mathematica
    t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k > n || k < 0, 0, t[n - 1, k - 1] + n*t[n - 2, k]]; Table[t[n, k], {n, 0, 10}, {k, 0, n}](* Clark Kimberling, Oct 19 2014 *)
(* Next, the polynomials *); z = 20; f[x_, n_] := x + n/f[x, n - 1]; f[x_, 0] = 1; t = Table[Factor[f[x, n]], {n, 0, z}]; u = Numerator[t]; TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]]  (* A249057 array *)
Flatten[CoefficientList[u, x]] (* A249057 sequence *)
(* Clark Kimberling, Oct 19 2014 *)

Formula

T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.
T(n,0) = A006882(n).
T(n+1,1) = A007911(n+3).
Sum_{k=0..n} T(n,k) = A000085(n+1).

Extensions

Corrected by Clark Kimberling, Oct 21 2014

A249074 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 4, 1, 6, 4, 1, 32, 14, 4, 1, 60, 72, 24, 4, 1, 384, 228, 120, 36, 4, 1, 840, 1392, 564, 176, 50, 4, 1, 6144, 4488, 3312, 1140, 240, 66, 4, 1, 15120, 31200, 14640, 6480, 2040, 312, 84, 4, 1, 122880, 104880, 97440, 37440, 11280, 3360, 392, 104, 4, 1, 332640
Offset: 0

Views

Author

Clark Kimberling, Oct 20 2014

Keywords

Comments

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + 2*(n+1)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A249075(n) for n >= 0.
(n-th term of column 1) = A087299(n) for n >= 1.

Examples

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (4 + x)/1, so that p(1,x) = 4 + x;
f(2,x) = (6 + 4*x + x^2)/(4 + x), so that p(2,x) = 6 + 4*x + x^2.
First 6 rows of the triangle of coefficients:
  1
  4    1
  6    4    1
  32   14   4    1
  60   72   24   4    1
  384  228  120  36   4   1

Links

Crossrefs

Cf. A249057, A249075, A087299.

Programs

  • Mathematica
    z = 11; p[x_, n_] := x + 2 n/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249074 array *)
Flatten[CoefficientList[u, x]] (* A249074 sequence *)
v = u /. x -> 1  (* A249075 *)
u /. x -> 0      (* A087299 *)
Showing 1-4 of 4 results.

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