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
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
Programs
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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
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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
Comments