A080850 Number triangle related to a problem of Knuth.

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 5, 6, 4, 1, 11, 10, 11, 10, 5, 1, 22, 21, 21, 21, 15, 6, 1, 43, 43, 42, 42, 36, 21, 7, 1, 85, 86, 85, 84, 78, 57, 28, 8, 1, 170, 171, 171, 169, 162, 135, 85, 36, 9, 1, 341, 341, 342, 340, 331, 297, 220, 121, 45, 10, 1, 683, 682, 683, 682, 671, 628, 517, 341, 166, 55, 11, 1

Offset

0,5

Comments

In lower-triangular form, the columns are the binomial transforms of the sequences with g.f. x^k/(1-x^3). The first three columns are A024493, A024494, A024495.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

T(n, 0) = A024493(n). T(n, k) = 0, k>n, T(n, n)=1. T(n, k) = T(n-1, k-1)+T(n-1, k).

G.f.: (1 - x)^3/((1 - 2*x)*(1 - (1 + y)*x)*(1 - x + x^2)). - Andrew Howroyd, Oct 01 2025

Example

Rows are:
  {1},
  {1,1},
  {1,2,1},
  {2,3,3,1},
  {5,5,6,4,1},
  {11,10,11,10,5,1},
  ...

Prog

(PARI) T(n)={my(u=Vec(1/(1-x^3) + O(x*x^n))); vector(n+1, n, vector(n, k, sum(i=k, n, binomial(n-1, i-1)*u[i-k+1])))}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Oct 01 2025

Crossrefs

Cf. A024493 (k=0), A024494 (k=1), A024495 (k=2).

Sequence in context: A091836 A291980 A238281 * A247453 A109449 A129570

Adjacent sequences: A080847 A080848 A080849 * A080851 A080852 A080853

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 20 2003

Extensions

Missing a(70) inserted by Sean A. Irvine, Oct 01 2025

Offset corrected by Andrew Howroyd, Oct 01 2025

Status

approved