A257813 G.f. satisfies: A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).

1, 0, 1, 0, 2, 0, 0, 4, 5, 0, 0, 8, 38, 14, 0, 0, 16, 184, 262, 42, 0, 0, 32, 720, 2460, 1602, 132, 0, 0, 64, 2480, 16360, 25837, 9260, 429, 0, 0, 128, 7840, 87920, 268134, 237870, 52040, 1430, 0, 0, 256, 23296, 408128, 2109040, 3638386, 2023992, 288494, 4862, 0, 0, 512, 66048, 1701504, 13676128, 40049492, 43815744, 16394336, 1590638, 16796, 0

Offset

0,5

Comments

The rightmost nonzero numbers in this triangle form the Catalan numbers (A000108).

Links

Table of n, a(n) for n=0..65.

Formula

G.f. A(x,y) satisfies: A(x/A(x,y)^2, y) = 1+x + (y-1)*x/A(x,y)^2.

Example

This triangle begins:
1;
0, 1;
0, 2, 0;
0, 4, 5, 0;
0, 8, 38, 14, 0;
0, 16, 184, 262, 42, 0;
0, 32, 720, 2460, 1602, 132, 0;
0, 64, 2480, 16360, 25837, 9260, 429, 0;
0, 128, 7840, 87920, 268134, 237870, 52040, 1430, 0;
0, 256, 23296, 408128, 2109040, 3638386, 2023992, 288494, 4862, 0;
0, 512, 66048, 1701504, 13676128, 40049492, 43815744, 16394336, 1590638, 16796, 0;
0, 1024, 180480, 6531840, 76845728, 349863976, 653001202, 487491424, 128720399, 8765044, 58786, 0;
0, 2048, 478720, 23485440, 386423488, 2571281744, 7476451420, 9591548748, 5139351752, 991185638, 48412190, 208012, 0; ...
Row sums (A120970) begin:
[1, 1, 2, 9, 60, 504, 4946, 54430, 655362, 8496454, 117311198, ...],
the g.f. of which satisfies: G(x) = 1 + Series_Reversion(x/G(x)^2).
GENERATING FUNCTION.
G.f.: A(x,y) = 1 + x*y + x^2*(2*y) + x^3*(4*y + 5*y^2) +
x^4*(8*y + 38*y^2 + 14*y^3) +
x^5*(16*y + 184*y^2 + 262*y^3 + 42*y^4) +
x^6*(32*y + 720*y^2 + 2460*y^3 + 1602*y^4 + 132*y^5) +
x^7*(64*y + 2480*y^2 + 16360*y^3 + 25837*y^4 + 9260*y^5 + 429*y^6) +
x^8*(128*y + 7840*y^2 + 87920*y^3 + 268134*y^4 + 237870*y^5 + 52040*y^6 + 1430*y^7) +
x^9*(256*y + 23296*y^2 + 408128*y^3 + 2109040*y^4 + 3638386*y^5 + 2023992*y^6 + 288494*y^7 + 4862*y^8) +...
where
A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).
RELATED SERIES.
A(x/A(x,y)^2, y) = 1 + y*x + (-2*y^2 + 2*y)*x^2 +
(3*y^3 - 7*y^2 + 4*y)*x^3 +
(-4*y^4 + 6*y^3 - 10*y^2 + 8*y)*x^4 +
(5*y^5 - 27*y^4 - 18*y^3 + 24*y^2 + 16*y)*x^5 +
(-6*y^6 - 14*y^5 - 312*y^4 + 60*y^3 + 240*y^2 + 32*y)*x^6 +
(7*y^7 - 147*y^6 - 1745*y^5 - 1675*y^4 + 2360*y^3 + 1136*y^2 + 64*y)*x^7 +
(-8*y^8 - 348*y^7 - 10744*y^6 - 25146*y^5 + 10246*y^4 + 21616*y^3 + 4256*y^2 + 128*y)*x^8 +
(9*y^9 - 1361*y^8 - 60738*y^7 - 267656*y^6 - 84094*y^5 + 265552*y^4 + 133952*y^3 + 14080*y^2 + 256*y)*x^9 +...

Prog

(PARI) {T(n, k) = local(A=[1]); for(i=1, n, A=Vec(1 + (y-1)*x + serreverse(x/Ser(A)^2))); polcoeff(A[n+1], k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A120970, A000108.

Sequence in context: A115179 A117434 A131742 * A278280 A351403 A213370

Adjacent sequences: A257810 A257811 A257812 * A257814 A257815 A257816

Keyword

nonn

Author

Paul D. Hanna, May 10 2015

Status

approved