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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, 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
0,3
COMMENTS
This triangle was found by George Plousos while exploring a variation of Aitken's array (A011971).
LINKS
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
(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(""))
CROSSREFS
Sequence in context: A032294 A146034 A374534 * A032029 A058571 A058492
KEYWORD
nonn,tabl
AUTHOR
George Plousos and Paul D. Hanna, Sep 22 2024
STATUS
approved