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 1-7 of 7 results.

A124889 Central terms of triangle A124328; a(n) = A124328(2n+1,n+1) for n>=0.

Original entry on oeis.org

1, 2, 9, 44, 235, 1296, 7329, 42104, 244719, 1434840, 8470517, 50280372, 299806572, 1794382548, 10773855210, 64865425760, 391457133672, 2367314649522, 14342441005900, 87035702613500, 528938145043707, 3218713327609648
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2006

Keywords

Crossrefs

Cf. A124328, A124889, A124890, A124891.

Formula

a(n) = (n+1)*A124890(n).

A127082 Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recursion: C_k(x) = ( 1 + Sum_{n>=1} x^n*C_{n-1+k}(x) )^(k+1).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 16, 28, 15, 4, 1, 64, 127, 85, 26, 5, 1, 308, 650, 531, 192, 40, 6, 1, 1728, 3737, 3600, 1551, 365, 57, 7, 1, 11046, 23996, 26266, 13416, 3635, 620, 77, 8, 1, 79065, 170866, 205353, 122770, 38556, 7356, 973, 100, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Comments

This is a variant of triangle A124328.

Examples

			C_k = [ 1 + x*C_k + x^2*C_{k+1} + x^3*C_{k+2} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^4;
C_2 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^3;
C_1 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^2;
C_0 = [ 1 + x*C_0 + x^2*C_1 + x^3*C_2 + x^4*C_3 +... ]^1;
thus the row sums equal column 0 shift left.
The triangle begins:
       1;
       1,       1;
       2,       2,       1;
       5,       7,       3,       1;
      16,      28,      15,       4,      1;
      64,     127,      85,      26,      5,     1;
     308,     650,     531,     192,     40,     6,     1;
    1728,    3737,    3600,    1551,    365,    57,     7,    1;
   11046,   23996,   26266,   13416,   3635,   620,    77,    8,   1;
   79065,  170866,  205353,  122770,  38556,  7356,   973,  100,   9,  1;
  625049, 1338578, 1716582, 1180496, 429515, 92730, 13412, 1440, 126, 10, 1;

Links

Crossrefs

Cf. variant: A124328;
Columns: A127083, A127084, A127085, A127086, A127090 (central terms).

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c, k, r}], {r, k, n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, k], {n,0,12}, {k, 0,n}]//Flatten (* G. C. Greubel, Jan 30 2020 *)
  • PARI
    {T(n,k)=if(n==k,1,polcoeff( (1 + x*sum(r=k,n-1,x^(r-k)*sum(c=k,r, T(r,c) ))+x*O(x^n))^(k+1),n-k))}

A124344 Number of ordered rooted trees on n nodes with thinning limbs.

Original entry on oeis.org

1, 1, 2, 4, 10, 25, 68, 187, 530, 1523, 4447, 13121, 39107, 117490, 355507, 1082234, 3312255, 10185125, 31450633, 97480337, 303157086, 945671951, 2958113722, 9276528602, 29158191215, 91845796986, 289874628176, 916536727561
Offset: 1

Views

Author

Christian G. Bower, Oct 30 2006, suggested by Franklin T. Adams-Watters

Keywords

Comments

A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Links

Crossrefs

Cf. A000108, A124343-A124348.
Row sums of A124328.

Formula

G.f.: A(x) = A0(x)+A1(x)+A2(x)+... where A0(x)=x, An(x) = x*(A0(x)+A1(x)+...+An(x))^n.

A127126 Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recurrence: C_k(x) = [ 1 + Sum_{n>=k+1} C_n(x)*x^(n-k) ]^(k+1) for k>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 77, 54, 18, 4, 1, 587, 412, 139, 30, 5, 1, 5484, 3834, 1314, 284, 45, 6, 1, 60582, 42131, 14658, 3217, 505, 63, 7, 1, 771261, 533558, 188012, 42100, 6680, 818, 84, 8, 1, 11102828, 7645065, 2721462, 621936, 100621, 12387, 1239, 108, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Comments

This is a variant of triangles: A127082, A124328.

Examples

			C_k = [ 1 + x*C_{k+1} + x^2*C_{k+2} + x^3*C_{k+3} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_4 + x^2*C_5 + x^3*C_6 + x^4*C_7 +... ]^4;
C_2 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^3;
C_1 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^2;
C_0 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^1.
The triangle begins:
         1;
         1,       1;
         3,       2,       1;
        13,       9,       3,      1;
        77,      54,      18,      4,      1;
       587,     412,     139,     30,      5,     1;
      5484,    3834,    1314,    284,     45,     6,    1;
     60582,   42131,   14658,   3217,    505,    63,    7,   1;
    771261,  533558,  188012,  42100,   6680,   818,   84,   8, 1;
  11102828, 7645065, 2721462, 621936, 100621, 12387, 1239, 108, 9, 1; ...

Links

Crossrefs

Columns: A127127, A127128, A127129, A127130.
Central terms: A127134.
Variants: A127082, A124328.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k-1)*Sum[T[r, c], {c,k+1,r}], {r,k+1,n}] +x^(n+1))^(k+1), x, n-k]]; Table[T[n, k], {n,0,12}, {k, 0,n}]//Flatten (* G. C. Greubel, Jan 27 2020 *)
  • PARI
    {T(n,k) = if(n==k,1,polcoeff( (1 + x*sum(r=k+1,n,x^(r-k-1)*sum(c=k+1,r, T(r,c))) +x*O(x^n))^(k+1),n-k))}

A124890 G.f.: A(x) = (1/x)*series_reversion(x^2/G124344(x)), where G124344(x) is the g.f. of A124344 and satisfies: A(x)^2 = (1/x)*G124344(x*A(x)).

Original entry on oeis.org

1, 1, 3, 11, 47, 216, 1047, 5263, 27191, 143484, 770047, 4190031, 23062044, 128170182, 718257014, 4054089110, 23026890216, 131517480529, 754865316100, 4351785130675, 25187530716367, 146305151254984, 852604651815745
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2006

Keywords

Comments

A124344(n) is the number of ordered rooted trees on n nodes with thinning limbs.

Crossrefs

Cf. A124344, A124328, A124889, A124891.

Formula

a(n) = A124889(n)/(n+1). a(n) = A124328(2n+1,n+1)/(n+1).

A124891 L.g.f.: A(x) = log(G124890(x)) where G124890(x) is the g.f. of A124890.

Original entry on oeis.org

1, 5, 25, 137, 766, 4379, 25355, 148273, 873574, 5177450, 30833342, 184355207, 1105977887, 6653964847, 40131748300, 242567280865, 1468928473132, 8910461020730, 54131814523902, 329299899410062, 2005674943792559
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2006

Keywords

Examples

			A(x) = x + 5*x^2/2 + 25*x^3/3 + 137*x^4/4 + 766*x^5/5 + 4379*x^6/6 +...
exp(A(x)) = G124890(x) where
G124890(x) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 216*x^5 + 1047*x^6 +...

Crossrefs

Cf. A124890, A124328, A124889.

Formula

a(n) = A124328(2n+2,n+1) for n>=0; thus a(n) is the number of ordered trees with 2(n+1) edges, with thinning limbs and with root of degree n+1.

A124329 Number of ordered trees with n edges, with thinning limbs and with root of degree 2. An ordered tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Original entry on oeis.org

0, 1, 2, 5, 10, 22, 46, 101, 220, 492, 1104, 2515, 5762, 13327, 30994, 72555, 170654, 403350, 957134, 2279947, 5449012, 13063595, 31406516, 75701507, 182902336, 442885682, 1074604288, 2612341855, 6361782006, 15518343596, 37912613630
Offset: 1

Views

Author

Emeric Deutsch, Nov 03 2006

Keywords

Comments

Column 2 of A124328.

Links

Crossrefs

Cf. A124344, A124328.

Programs

  • Maple
    G:=(1-z-2*z^2-sqrt(1-2*z-3*z^2+4*z^3))/2/z^2/(1-z): Gser:=series(G,z=0,40): seq(coeff(Gser,z,n),n=1..36);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [0$2, 1, 2][n+1],
      ((3*n+3)*a(n-1) +(n-4)*a(n-2) -(7*n-13)*a(n-3)
       +(4*n-10)*a(n-4)) / (n+2))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 08 2014
  • Mathematica
    Rest[CoefficientList[Series[(1-x-2*x^2-Sqrt[1-2*x-3*x^2+4*x^3])/2/x^2/(1-x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Sep 04 2014 *)
Table[2*Sum[((Binomial[2*k + 1, k + 1]*Binomial[n - k, k + 1])/(k + 2)), {k, 0, (n - 1)/2}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 22 2016, after Vladimir Kruchinin *)
  • Maxima
    a(n):=2*sum((binomial(2*k+1, k+1)*binomial(n-k, k+1))/(k+2), k, 0, (n-1)/2); /* Vladimir Kruchinin, Apr 21 2016 */

Formula

G.f.: [1-z-2z^2-sqrt(1-2z-3z^2+4z^3)]/[2(1-z)z^2].
a(n) ~ sqrt(493+101*sqrt(17)) * (1+sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(n+7/2)). - Vaclav Kotesovec, Sep 04 2014
a(n) = 2*Sum_{k = 0..(n-1)/2} binomial(2*k+1, k+1)*binomial(n-k, k+1)/(k+2). - Vladimir Kruchinin, Apr 21 2016
D-finite with recurrence (n+2)*a(n) +3*(-n-1)*a(n-1) +(-n+4)*a(n-2) +(7*n-13)*a(n-3) +2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.