A353133 Coefficients of expansion of f(x) = (1+x*m(x))^5*(x^2*(x*m(x))'+1) where m(x) is the generating function for A001006.

1, 5, 16, 47, 136, 392, 1130, 3262, 9434, 27337, 79364, 230815, 672380, 1961635, 5730860, 16763685, 49093260, 143924943, 422352816, 1240529133, 3646710456, 10728322770, 31584554610, 93048320820, 274292367650, 809044988695, 2387642856380, 7050001551361, 20826624824612, 61552574382856

Offset

0,2

Comments

2*x^7*f(x) is the generating function for the number of Dyck paths with L(D)=7 where L(D) is the product of binomial coefficients (u_i(D)+d_i(D) choose u_i(D)), where u_i(D) is the number of up-steps between the i-th and (i+1)-st down step and d_i(D) is the number of down-steps between the i-th and (i+1)-st up step.

Links

Table of n, a(n) for n=0..29.

Kassie Archer and Christina Graves, Pattern-restricted permutations composed of 3-cycles, arXiv:2104.12664 [math.CO], 2021.

Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.

Prog

(PARI) m(x) = (1-x-sqrt(1-2*x-3*(x^2)))/(2*(x^2));
my(x='x+O('x^30)); Vec((1+x*m(x))^5*(x^2*(x*m(x))'+1)) \\ Michel Marcus, Apr 25 2022

Crossrefs

Cf. A001006, A005717, A005773, A025565, A225034.

Sequence in context: A086749 A194541 A094586 * A140336 A197201 A166868

Adjacent sequences: A353130 A353131 A353132 * A353134 A353135 A353136

Keyword

nonn

Author

Kassie Archer, Apr 25 2022

Status

approved