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A185375
a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90.
2
0, 0, 1, 91, 966, 5082, 18447, 53053, 129948, 282948, 562989, 1043119, 1824130, 3040830, 4868955, 7532721, 11313016, 16556232, 23683737, 33201987, 45713278, 61927138, 82672359, 108909669, 141745044, 182443660
OFFSET
0,4
COMMENTS
Third column (k=2) of A008958.
FORMULA
a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90.
a(n) = binomial(2*n+1,5)*(10*n-17)/3.
From G. C. Greubel, Jun 28 2017: (Start)
G.f.: x^2*(1 + 84*x + 350*x^2 + 196*x^3 + 9*x^4)/(1 - x)^7.
E.g.f.: (1/90)*x^2*(45 + 1320 x + 2280 x^2 + 864 x^3 + 80 x^4)*exp(x). (End)
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, Apr 23 2021
a(n) = Sum_{1 <= i <= j <= n-1} (2*i - 1)^2 * (2*j - 1)^2. - Peter Bala, Sep 03 2023
MATHEMATICA
Table[n*(n - 1)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(10*n - 17)/90, {n, 0, 50}] (* G. C. Greubel, Jun 28 2017 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 1, 91, 966, 5082, 18447}, 30] (* Harvey P. Dale, Oct 10 2021 *)
PROG
(PARI) a(n) = binomial(2*n+1, 5)*(10*n-17)/3 \\ Michel Marcus, Jun 18 2013
(Magma) [n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90 : n in [0..50]]; // Wesley Ivan Hurt, Apr 23 2021
CROSSREFS
Third column (k=2) of A008958 Triangle of central factorial numbers.
Cf. A103220.
Sequence in context: A129255 A093291 A254519 * A036526 A170955 A020316
KEYWORD
easy,nonn
AUTHOR
Wesley Transue, Jan 21 2012
STATUS
approved