A111373 A generalized Pascal triangle.

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 1, 3, 0, 0, 4, 0, 0, 1, 0, 7, 0, 0, 5, 0, 0, 1, 0, 0, 12, 0, 0, 6, 0, 0, 1, 12, 0, 0, 18, 0, 0, 7, 0, 0, 1, 0, 30, 0, 0, 25, 0, 0, 8, 0, 0, 1, 0, 0, 55, 0, 0, 33, 0, 0, 9, 0, 0, 1, 55, 0, 0, 88, 0, 0, 42, 0, 0, 10, 0, 0, 1, 0, 143, 0, 0, 130, 0, 0, 52, 0, 0, 11, 0, 0, 1

Offset: 0

Author

N. J. A. Sloane, Nov 09 2005

Keywords

• nonn
• easy
• tabl

Comments

First diagonal is A000012, the all 1's sequence. Second nonzero diagonal is A000027 = n. Third nonzero diagonal is A027379 = n*(n+5)/2 for n>=1, or essentially A000217(n) - 3. Fourth nonzero diagonal is A111396. - Jonathan Vos Post, Nov 10 2005

Row sums are A126042. - Paul Barry, Dec 16 2006

Examples

			Triangle begins:
   1;
   0,  1;
   0,  0,  1;
   1,  0,  0,  1;
   0,  2,  0,  0,  1;
   0,  0,  3,  0,  0,  1;
   3,  0,  0,  4,  0,  0,  1;
   0,  7,  0,  0,  5,  0,  0,  1;
   0,  0, 12,  0,  0,  6,  0,  0,  1;
  12,  0,  0, 18,  0,  0,  7,  0,  0,  1;
   0, 30,  0,  0, 25,  0,  0,  8,  0,  0,  1;
   0,  0, 55,  0,  0, 33,  0,  0,  9,  0,  0,  1;
  55,  0,  0, 88,  0,  0, 42,  0,  0, 10,  0,  0,  1;
Production matrix is
  0, 1;
  0, 0, 1;
  1, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1, 0, 0, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.
• Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 7, 10.
• Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

Crossrefs

First column is A001764. Bears same relation to A001764 as A053121 does to A000108.

Cf. A000012, A000027, A027379, A111396, A126042 (row sums), A167543.

Programs

Magma

function A111373(n,k)
  if k eq n then return 1;
  elif ((n-k) mod 3) eq 0 then return (3/(n-k))*Binomial(n, Floor((n-k-3)/3))*(k+1);
  else return 0;
  end if; return A111373;
end function;
[A111373(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 30 2022

Mathematica

T[n_, k_]= If[k==n, 1, If[Mod[n-k, 3]==0, (3/(n-k))*Binomial[n, (n-k)/3 -1]*(k + 1), 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 30 2022 *)

SageMath

def A111373(n,k):
    if(k==n): return 1
    elif ((n-k)%3==0): return (3/(n-k))*binomial(n, (n-k)/3 -1)*(k+1)
    else: return 0
flatten([[A111373(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 30 2022

Formula

Each term is the sum of the two terms above it to the left and two steps to the right.

From Paul Barry, Dec 16 2006: (Start)

Riordan array (g(x^3),x*g(x^3)) where g(x)=(2/sqrt(3x))*sin(asin(sqrt(27x/4))/3), the g.f. of A001764;

Number triangle T(n,k) = C(3*floor((n+2k)/3)-2k,floor((n+2k)/3)-k)*(k+1)/(2*floor((n+2k)/3)-k+ 1)(2*cos(2*pi*(n-k)/3)+1)/3. (End)

Inverse of Riordan array (1/(1+x^3), x/(1+x^3)), A126030. - Paul Barry, Dec 16 2006

G.f. (x*A(x))^k=sum{n>=k, T(n,k)*x^n}, where A(x)=1+x^3*A(x)^3. - Vladimir Kruchinin, Feb 18 2011

From G. C. Greubel, Jul 30 2022: (Start)

T(n, k) = (3/(n-k))*binomial(n, (n-k)/3 -1)*(k+1) for ( (n-k) mod 3 ) = 0, otherwise 0, with T(n, n) = 1.

T(n, n-3) = A000027(n-2), n >= 3.

T(n, n-6) = A027379(n-5), n >= 6.

T(n, n-9) = A111396(n-8), n >= 9.

T(n, n-12) = A167543(n+5), n >= 12.

Sum_{k=0..n} T(n, k) = A126042(n). (End)

Extensions

More terms from Kerri Sullivan (ksulliva(AT)ashland.edu), Jan 23 2006