A136304 - OEIS

A136304

Expansion of g.f. (1-z)^2*(1-sqrt(1-4*z))/(2*z*(1 - 3*z + 2*z^2 - z^3)).

1, 2, 5, 14, 40, 116, 344, 1047, 3273, 10500, 34503, 115838, 396244, 1377221, 4851665, 17285662, 62173297, 225424527, 822919439, 3021713140, 11151957809, 41340655956, 153853915410, 574593145517, 2152679745351, 8087904580883, 30466311814036, 115036597198845

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OFFSET

0,2

COMMENTS

Previous name was: Transform of A000108 by the T_{0,0} transformation (see link).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Richard Choulet, Curtz-like transformation.

FORMULA

G.f.: (1-z)^2*(1-sqrt(1-4*z))/(2*z*(1 - 3*z + 2*z^2 - z^3)).

Conjecture: (n+1)*a(n) + (-8*n+1)*a(n-1) + 3*(7*n-8)*a(n-2) + (-23*n+49)*a(n-3) + (13*n-32)*a(n-4) + 2*(-2*n+7)*a(n-5) = 0. - R. J. Mathar, Feb 29 2016

Conjecture verified using differential equation -z^3 + 3*z^2 - 3*z + 1 + (6*z^5 - 20*z^4 + 20*z^3 - 18*z^2 + 7*z - 1)*g(z) + (4*z^6 - 13*z^5 + 23*z^4 - 21*z^3 + 8*z^2 - z)*g'(z) satisfied by the G.f. - Robert Israel, Feb 23 2026

a(n) = Sum_{j=0..n} A034943(j+1)*A000108(n-j). - G. C. Greubel, Apr 19 2021

MAPLE

f:= gfun:-rectoproc({(6 + 4*n)*a(n) + (-33 - 13*n)*a(n + 1) + (66 + 23*n)*a(n + 2) + (-81 - 21*n)*a(n + 3) + (39 + 8*n)*a(n + 4) + (-6 - n)*a(n + 5), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 14, a(4) = 40}, a(n), remember):

map(f, [$0..40]); # Robert Israel, Feb 23 2026

MATHEMATICA

A034943[n_]:= A034943[n]= Sum[Binomial[n+k-1, 3*k], {k, 0, n/2}];

Table[Sum[A034943[j+1]*CatalanNumber[n-j], {j, 0, n}], {n, 0, 35}] (* G. C. Greubel, Apr 19 2021 *)

PROG

(Magma)

A034943:= func< n | (&+[Binomial(n+j-1, 3*j): j in [0..Floor(n/2)]]) >;

[(&+[A034943(j+1)*Catalan(n-j): j in [0..n]]): n in [0..35]]; // G. C. Greubel, Apr 19 2021

(SageMath)

def A034943(n): return sum(binomial(n+j-1, 3*j) for j in (0..n//2))

[sum(A034943(j+1)*catalan_number(n-j) for j in (0..n)) for n in (0..35)] # G. C. Greubel, Apr 19 2021

CROSSREFS