A120019 Square table, read by antidiagonals, of self-compositions of A120010.

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 9, 10, 6, 1, 5, 16, 30, 32, 18, 1, 6, 25, 68, 114, 116, 53, 1, 7, 36, 130, 312, 480, 440, 158, 1, 8, 49, 222, 710, 1536, 2157, 1708, 481, 1, 9, 64, 350, 1416, 4070, 8000, 10092, 6760, 1491, 1, 10, 81, 520, 2562, 9348, 24365, 43472, 48525

Offset

1,5

Comments

The g.f. of row n is the composition: (1-sqrt(1-4*x))/2 o x/(1-nx) o (x-x^2).

Links

Table of n, a(n) for n=1..64.

Formula

T(n, k) = Sum_{j=1..k}Catalan(k-j)*[Sum_{i=1..j}(-1)^(j-i)*n^(i-1)*C(k-j+i, j-i)*C(k-j+i-1, i-1)]; Also, T(n, k) = Sum_{j=0..k-1}n^j*[Sum_{i=j..k-1}(-1)^(i-j)*Catalan(k-i-1)*C(k-i+j, i-j)*C(k-i+j-1, j)]; where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

Example

Square table begins:
1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, ...
1, 2, 4, 10, 32, 116, 440, 1708, 6760, 27232, ...
1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, ...
1, 4, 16, 68, 312, 1536, 8000, 43472, 243808, 1400448, ...
1, 5, 25, 130, 710, 4070, 24365, 151330, 968785, 6355795, ...
1, 6, 36, 222, 1416, 9348, 63768, 448188, 3234216, 23875296, ...
1, 7, 49, 350, 2562, 19236, 148085, 1167488, 9409645, 77367087, ...
1, 8, 64, 520, 4304, 36320, 312512, 2740672, 24476800, 222358528, ...
1, 9, 81, 738, 6822, 64026, 610245, 5906502, 58033953, 578488563, ...
1, 10, 100, 1010, 10320, 106740, 1117880, 11855660, 127313320, ...
Successive self-compositions of F(x), the g.f. of A120010, begin:
F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + 3x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 68x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + 710x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + 9348x^6 +...

Prog

(PARI) {T(n, k)=sum(j=1, k, binomial(2*k-2*j, k-j)/(k-j+1)* sum(i=1, j, (-1)^(j-i)*binomial(k-j+i, j-i)*binomial(k-j+i-1, i-1)*n^(i-1)))}

Crossrefs

Rows: A120010, A120017, A120018; Diagonals: A120020, A120021. Variant: A120013.

Sequence in context: A193736 A292975 A056863 * A145034 A286510 A336842

Adjacent sequences: A120016 A120017 A120018 * A120020 A120021 A120022

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 14 2006

Status

approved