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A349769
a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(k,floor(k/2)).
0
1, 2, 7, 31, 143, 686, 3417, 17382, 89791, 470134, 2487593, 13273921, 71341921, 385786298, 2097096263, 11451611919, 62784274623, 345436875758, 1906568766489, 10552637998329, 58556298508449, 325676578717698, 1815140080977303, 10135993961893674
OFFSET
0,2
FORMULA
From Vaclav Kotesovec, Nov 29 2021: (Start)
D-finite recurrence: n*(n+1)^2*(32*n^4 - 208*n^3 + 456*n^2 - 380*n + 81)*a(n) = 2*n*(64*n^6 - 320*n^5 + 344*n^4 + 288*n^3 - 332*n^2 - 363*n + 243)*a(n-1) + 2*(n-1)*(32*n^6 - 80*n^5 - 624*n^4 + 2492*n^3 - 2257*n^2 - 596*n + 927)*a(n-2) + 2*(n-2)*(832*n^6 - 6272*n^5 + 16360*n^4 - 15456*n^3 - 1016*n^2 + 8375*n - 3195)*a(n-3) - 9*(n-3)^2*(n-2)*(32*n^4 - 80*n^3 + 24*n^2 + 36*n - 19)*a(n-4).
a(n) ~ (1 + sqrt(2))^(2*n + 3/2) / (2*Pi*n). (End)
MATHEMATICA
Table[Sum[Binomial[n, k]^2 Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 22}]
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)^2 * binomial(k, k\2)); \\ Michel Marcus, Nov 29 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Nov 29 2021
STATUS
approved