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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

%I #10 Oct 04 2024 14:21:07

%S 1,1,3,3,7,17,17,37,81,179,179,375,787,1655,3489,3489,7157,14689,

%T 30165,61985,127459,127459,258407,523971,1062631,2155427,4372839,

%U 8873137,8873137,17873733,36005873,72535717,146134065,294423557,593219953,1195313043,1195313043,2399499223,4816872179,9669750231,19412036179,38970206423,78234836403,157062892759,315321098561,315321098561

%N 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.

%C This triangle was found by _George Plousos_ while exploring a variation of Aitken's array (A011971).

%H Paul D. Hanna, <a href="/A376177/b376177.txt">Table of n, a(n) for n = 0..1275</a>

%F 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.

%F 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.

%F 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_.

%e 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 + ...

%e which is defined by A(x,y) = (B(x) - 2*(B(x*y) - 1)/x) / (1 - (2+x)*y),

%e where B(x) = 1 + x*B( 2*x/(1-x) )/(1-x) is the g.f. B(x) for A126443,

%e 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 + ...

%e This triangle begins

%e 1,

%e 1, 3,

%e 3, 7, 17,

%e 17, 37, 81, 179,

%e 179, 375, 787, 1655, 3489,

%e 3489, 7157, 14689, 30165, 61985, 127459,

%e 127459, 258407, 523971, 1062631, 2155427, 4372839, 8873137,

%e 8873137, 17873733, 36005873, 72535717, 146134065, 294423557, 593219953, 1195313043,

%e ...

%o (PARI) {A126443(n) = if(n==0, 1, sum(k=0, n-1, binomial(n-1, k) * 2^k * A126443(k)))}

%o {T(n,k) = sum(j=0,k, binomial(k,j) * 2^j * A126443(n-k+j) )}

%o for(n=0,10, for(k=0,n, print1(T(n,k),", ")); print(""))

%Y Cf. A011971, A126443.

%K nonn,tabl

%O 0,3

%A _George Plousos_ and _Paul D. Hanna_, Sep 22 2024