1,5

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

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).

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 +...

(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)))}

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

Sequence in context: A089940 A123974 A056863 this_sequence A159933 A128314 A025564

Adjacent sequences: A120016 A120017 A120018 this_sequence A120020 A120021 A120022

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 14 2006