OFFSET
0,3
COMMENTS
This triangle was found by George Plousos while exploring a variation of Aitken's array (A011971).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1275
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.
EXAMPLE
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,
...
PROG
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
George Plousos and Paul D. Hanna, Sep 22 2024
STATUS
approved