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.

A168592 G.f.: exp( Sum_{n>=1} A082758(n)*x^n/n ), where A082758(n) = sum of the squares of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 14, 80, 509, 3459, 24579, 180389, 1356743, 10402493, 81004516, 638886082, 5093081983, 40971735401, 332187974718, 2711668091448, 22267979870143, 183830653156341, 1524747465249750, 12700172705956876, 106187411693668179
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2009

Keywords

Comments

Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 0 and two kinds of (1,1). - Alois P. Heinz, Oct 07 2015
Number of pairs of noncrossing paths of length n which start and end together, each taking steps (1,0), (1,1) or (1,-1) (i.e., Motzkin-type). - Nicholas R. Beaton, Jun 17 2024

Examples

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 509*x^4 + 3459*x^5 + ...
log(A(x)) = 3*x + 19*x^2/2 + 141*x^3/3 + 1107*x^4/4 + 8953*x^5/5 + ... + A082758(n)*x^n/n + ...

Links

Crossrefs

Cf. A168590, A168593, A082758, A027907, A168595, A218321, A263316.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
      add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 3, 14, 80][n+1],
      ((10*(n+1))*(16*n^3-20*n^2-n-1) *a(n-1)
      +(-944*n^4+2596*n^3-1924*n^2+236*n+30) *a(n-2)
      +(90*(n-2))*(16*n^3-52*n^2+45*n-6) *a(n-3)
      -(81*(2*n-5))*(n-2)*(n-3)*(4*n-1) *a(n-4))/
      ((n+1)*(4*n-5)*(2*n+1)*(n+2)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015
  • Mathematica
    (1/x)*InverseSeries[x*(1 - x)^2/((1 + x)^2*(1 - x + x^2)) + O[x]^30, x] // CoefficientList[#, x]& (* Jean-François Alcover, Jun 09 2018 *)
  • PARI
    {a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sum(k=0,2*m, polcoeff((1+x+x^2)^m,k)^2)*x^m/m) +x*O(x^n)),n))}
  • PARI
    {a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))),n)}

Formula

G.f.: A(x) = (1/x)*Series_Reversion[x*(1-x)^2/((1+x)^2*(1-x+x^2))].
G.f.: A(x) satisfies A(x^2) = M(x)*M(-x), where M(x) is the g.f. of A001006. - Alexander Burstein, Oct 03 2017
G.f.: A(x) satisfies A(x^2) = (1-x - sqrt(1-2*x-3*x^2))*(1+x - sqrt(1+2*x-3*x^2))/(4*x^4). - Paul D. Hanna, Oct 05 2017, concluded from formula of Alexander Burstein.

A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.

Original entry on oeis.org

1, 2, 18, 140, 1330, 12852, 130284, 1348776, 14247090, 152618180, 1654120468, 18096447096, 199536967084, 2214714164600, 24720932068200, 277289164574640, 3123590583844530, 35318969120870820, 400692715550057700, 4559427798654821400, 52020436064931914580
Offset: 0

Views

Author

Paul D. Hanna, Sep 08 2012

Keywords

Examples

			L.g.f.: L(x) = 2*x + 18*x^2/2 + 140*x^3/3 + 1330*x^4/4 + 12852*x^5/5 + 130284*x^6/6 + ...
where
exp(L(x)) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 + ... + A216585(n)*x^n/n + ...
The central trinomial coefficients (A002426) begin:
[1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, ...];
The central binomial coefficients (A000984) begin:
[1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, ...].

Links

Crossrefs

Cf. A216585, A151341, A168595, A002426, A000984.

Programs

  • Mathematica
    Table[Binomial[2*n, n]*Sum[ Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Feb 27 2017 *)
  • PARI
    {a(n) = polcoeff((1+x+x^2)^n,n) * polcoeff((1+2*x+x^2)^n,n)}
  • PARI
    {a(n)=binomial(2*n,n)*sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))}
for(n=0,21,print1(a(n),", "))

Formula

a(n) = binomial(2*n, n) * Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k).
Logarithmic derivative of A216585, after ignoring initial term a(0).
a(n) = [x^n*y^n] ( 1 + (x + y)^2 + (x + y)^4 )^n. - Peter Bala, Feb 17 2020
G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+4*x))/sqrt(1+4*x). - Mark van Hoeij, May 13 2025

A253255 G.f. satisfies: A(x) = (1 - x^3*A(x)^3)^2 / (1 - x*A(x))^4.

Original entry on oeis.org

1, 4, 26, 202, 1731, 15780, 150117, 1473292, 14807363, 151638550, 1576616125, 16598802248, 176599380271, 1895767748376, 20508188211018, 223348309510194, 2446792909432683, 26944972018189698, 298111489130625351, 3312016395569631402, 36935315970911333184, 413308467174788509668
Offset: 0

Views

Author

Paul D. Hanna, May 31 2015

Keywords

Comments

Self-convolution of A253256.

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 202*x^3 + 1731*x^4 + 15780*x^5 +...
where A(x) = (1 - x^3*A(x)^3)^2 / (1 - x*A(x))^4.
The logarithm begins:
log(A(x)) = 4*x + 36*x^2/2 + 358*x^3/3 + 3748*x^4/4 + 40404*x^5/5 + 443886*x^6/6 + 4941654*x^7/7 +...+ A168595(n)*x^n/n +...

Crossrefs

Cf. A168595, A106228, A064641, A253256.

Programs

  • PARI
    {a(n) = local(A=1); A = (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}
{a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)*x^k/k) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} A168595(n)*x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).
(2) A(x) = (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ).

A253256 G.f. satisfies: A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.

Original entry on oeis.org

1, 2, 11, 79, 647, 5727, 53367, 515802, 5123303, 51977485, 536320688, 5610909773, 59379328267, 634538481389, 6837466955193, 74210071037031, 810527496757335, 8901979424068377, 98253966680382102, 1089260346498608721, 12123804391067414676, 135427509933882292680, 1517725698030921469890
Offset: 0

Views

Author

Paul D. Hanna, May 31 2015

Keywords

Comments

Self-convolution yields A253255.

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 79*x^3 + 647*x^4 + 5727*x^5 + 53367*x^6 +...
where A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.
The logarithm begins:
log(A(x)) = 2*x + 18*x^2 + 179*x^3 + 1874*x^4 + 20202*x^5 + 221943*x^6 + 2470827*x^7/7 +...+ A168595(n)/2*x^n/n +...

Crossrefs

Cf. A168595, A106228, A064641, A253255.

Programs

  • PARI
    {a(n) = local(A=1); A = sqrt( (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}
{a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)/2 * x^k/k) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} A168595(n)/2 * x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).
(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ) ).
(3) A(x) = sqrt( (1-x*A(x) - sqrt(1 - 6*x*A(x) - 3*x^2*A(x)^2)) / (2*x*(1+x*A(x))) ).
Showing 1-4 of 4 results.

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

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