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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.
0
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
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
KEYWORD
nonn
AUTHOR
Kassie Archer, Apr 25 2022
STATUS
approved