A121372 Triangle, read by rows of length A003056(n) for n >= 1, defined by the recurrence: T(n,k) = T(n-k,k-1) - T(n-k,k) for n > k > 1, with T(n,1) =(-1)^(n-1) for n >= 1.

1, -1, 1, 1, -1, -1, 1, 0, -1, 0, 1, 1, 1, -1, -1, -1, 0, 1, 0, -1, -1, 0, 2, 1, 1, 1, -1, -1, -1, -1, 1, 0, 1, 0, -2, -1, -1, 0, 2, 1, 1, 1, -2, 0, 1, -1, -1, 2, 1, -1, 1, 0, -2, -1, 0, -1, 0, 3, 1, -1, 1, 1, -3, -2, 1, -1, -1, 2, 1, -1, 1, 0, -3, -1, 2, 1, -1, 0, 4, 2, -1, -1, 1, 1, -3, -1, 2, 0, -1, -1, 3, 1, -3, -1, 1, 0, -4, -2, 2, 1, -1, 0, 4, 2, -3

Offset: 1

Author

Paul D. Hanna, Jul 24 2006

Keywords

• sign
• tabf

Comments

Row sums equal A003406 (offset 1), the expansion of Ramanujan's function: R(x) = 1 + Sum_{n>=1} (x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n))).

Examples

			Triangle begins:
   1;
  -1;
   1,  1;
  -1, -1;
   1,  0;
  -1,  0,  1;
   1,  1, -1;
  -1, -1,  0;
   1,  0, -1;
  -1,  0,  2,  1;
   1,  1, -1, -1;
  -1, -1,  1,  0;
   1,  0, -2, -1;
  -1,  0,  2,  1;
   1,  1, -2,  0,  1;
  -1, -1,  2,  1, -1;
   1,  0, -2, -1,  0;
  -1,  0,  3,  1, -1;
   1,  1, -3, -2,  1;
  -1, -1,  2,  1, -1;
   1,  0, -3, -1,  2,  1;
  -1,  0,  4,  2, -1, -1;
   1,  1, -3, -1,  2,  0;
  -1, -1,  3,  1, -3, -1;
   1,  0, -4, -2,  2,  1;
  -1,  0,  4,  2, -3, -1;
   1,  1, -4, -2,  3,  1;
  -1, -1,  4,  2, -3,  0,  1;
   1,  0, -4, -2,  4,  2, -1;
  -1,  0,  5,  2, -4, -2,  0;
   1,  1, -5, -2,  5,  1, -1;
  -1, -1,  4,  2, -5, -2,  1;
   1,  0, -5, -2,  5,  2, -1;
  -1,  0,  6,  3, -6, -3,  1;
   1,  1, -5, -3,  6,  2, -1;
  -1, -1,  5,  2, -7, -2,  3,  1;
  ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..10075

Crossrefs

Cf. A003406, A008289.

Programs

PARI

{T(n, k)=if(n
				

PARI

/* Using generating formula for columns */
{tr(n) = floor((sqrt(1+8*n)-1)/2)} \\ number of terms in row n
{T(n,k) = polcoeff( x^(k*(k+1)/2) / prod(j=1,k, 1 + x^j +x*O(x^n)), n)}
{for(n=1,50, for(k=1, tr(n), print1(T(n,k),", "));print(""))} \\ Paul D. Hanna, Jan 28 2024

Formula

G.f. of column k: x^(k*(k+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^k)) for k >= 1.