A168593 -id:A168593 - cp's OEIS frontend

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.

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

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

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

			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 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

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

(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 *)

{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))}

{a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))),n)}

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.

A168590 G.f.: exp( Sum_{n>=1} A168591(n)*x^n/n ), where A168591(n) = sum of the n-th power of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 14, 310, 71399, 153056789, 2826352872319, 445742192193898313, 602479884829000885595175, 7000510736697461064666950774905, 701725717683874683612335083605682943282

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 310*x^3 + 71399*x^4 +...
log(A(x)) = 3*x + 19*x^2/2 + 831*x^3/3 + 281907*x^4/4 +...+ A168591(n)*x^n/n +...

Cf. A168591, A168592, A168593, A027907.

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sum(k=0,2*m,polcoeff((1+x+x^2)^m,k)^m)*x^m/m) +x*O(x^n)),n))}

A251686 G.f.: exp( Sum_{n>=1} A132303(n)/3 * x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 1, 8, 100, 1556, 27260, 515510, 10284094, 213433728, 4566363088, 100082133066, 2236952393302, 50817223209451, 1170319824912699, 27268900054818390, 641812268110993694, 15239341125950643462, 364655982858022960206, 8785745372509009963892, 212976842702489760621536

Offset: 0

Paul D. Hanna, Feb 28 2015

nonn

Self-convolution cube yields A168593.

			G.f.: A(x) = 1 + x + 8*x^2 + 100*x^3 + 1556*x^4 + 27260*x^5 +...
where
log(A(x)) = 1*x + 15*x^2/2 + 277*x^3/3 + 5727*x^4/4 + 125301*x^5/5 + 2843643*x^6/6 + 66214485*x^7/7 + 1571497119*x^8/8 +...+ A132303(n)/3*x^n/n +...

Cf. A168593, A132303, A027907.

{A027907(n,k) = polcoeff((1+x+x^2)^n, k)}
{A132303(n) = sum(k=0, 2*n, A027907(n,k)^3)}
{a(n) = if(n==0, 1, polcoeff(exp(sum(m=1, n, A132303(m)/3 * x^m/m) +x*O(x^n)), n))}
for(n=0,30,print1(a(n),", "))

cp's OEIS Frontend

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.

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A168590 G.f.: exp( Sum_{n>=1} A168591(n)*x^n/n ), where A168591(n) = sum of the n-th power of the trinomial coefficients in row n of triangle A027907.

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A251686 G.f.: exp( Sum_{n>=1} A132303(n)/3 * x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.

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