A376177 Triangle, read by rows, where T(n,k) = T(n-1,k-1) + 2*T(n,k-1) when k > 0, else T(n,0) = T(n-1,n-1) when n > 0, with T(0,0) = 1.

1, 1, 3, 3, 7, 17, 17, 37, 81, 179, 179, 375, 787, 1655, 3489, 3489, 7157, 14689, 30165, 61985, 127459, 127459, 258407, 523971, 1062631, 2155427, 4372839, 8873137, 8873137, 17873733, 36005873, 72535717, 146134065, 294423557, 593219953, 1195313043, 1195313043, 2399499223, 4816872179, 9669750231, 19412036179, 38970206423, 78234836403, 157062892759, 315321098561, 315321098561

Offset: 0

Author

George Plousos and Paul D. Hanna, Sep 22 2024

Keywords

• nonn
• tabl

Comments

This triangle was found by George Plousos while exploring a variation of Aitken's array (A011971).

Examples

			G.f.: A(x,y) = 1 + (3*y + 1)*x + (17*y^2 + 7*y + 3)*x^2 + (179*y^3 + 81*y^2 + 37*y + 17)*x^3 + (3489*y^4 + 1655*y^3 + 787*y^2 + 375*y + 179)*x^4 + (127459*y^5 + 61985*y^4 + 30165*y^3 + 14689*y^2 + 7157*y + 3489)*x^5 + (8873137*y^6 + 4372839*y^5 + 2155427*y^4 + 1062631*y^3 + 523971*y^2 + 258407*y + 127459)*x^6 + ...
which is defined by A(x,y) = (B(x) - 2*(B(x*y) - 1)/x) / (1 - (2+x)*y),
where B(x) = 1 + x*B( 2*x/(1-x) )/(1-x) is the g.f. B(x) for A126443,
B(x) = 1 + x + 3*x^2 + 17*x^3 + 179*x^4 + 3489*x^5 + 127459*x^6 + 8873137*x^7 + 1195313043*x^8 + 315321098561*x^9 + ... + A126443(n)*x^n + ...
This triangle begins
  1,
  1, 3,
  3, 7, 17,
  17, 37, 81, 179,
  179, 375, 787, 1655, 3489,
  3489, 7157, 14689, 30165, 61985, 127459,
  127459, 258407, 523971, 1062631, 2155427, 4372839, 8873137,
  8873137, 17873733, 36005873, 72535717, 146134065, 294423557, 593219953, 1195313043,
  ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1275

Crossrefs

Cf. A011971, A126443.

Programs

PARI

{A126443(n) = if(n==0, 1, sum(k=0, n-1, binomial(n-1, k) * 2^k * A126443(k)))}
{T(n,k) = sum(j=0,k, binomial(k,j) * 2^j * A126443(n-k+j) )}
for(n=0,10, for(k=0,n, print1(T(n,k),", ")); print(""))

Formula

If k > 0, T(n,k) = T(n-1,k-1) + 2*T(n,k-1), else if n > 0, T(n,0) = T(n-1,n-1), with T(0,0) = 1.

T(n,k) = Sum_{j=0..k} binomial(k,j) * 2^j * A126443(n-k+j), where A126443(m) = Sum_{k=0..m-1} binomial(m-1, k) * 2^k * A126443(k) for m > 0 with A126443(0) = 1.

G.f. A(x,y) = (B(x) - 2*(B(x*y) - 1)/x) / (1 - (2+x)*y), where B(x) = 1 + x*B( 2*x/(1-x) )/(1-x) is the g.f. B(x) for A126443 given therein by Ilya Gutkovskiy.