A395614 a(n) = Sum_{k=0..floor(n/4)} binomial(n,k) * binomial(2*n,n+4*k).

1, 2, 6, 20, 74, 302, 1320, 5980, 27458, 126380, 580906, 2666116, 12231344, 56158026, 258261072, 1190133320, 5496101250, 25431995956, 117890399940, 547329545688, 2544504863974, 11843220123454, 55181363382360, 257352009473648, 1201269305383568, 5611796083011552

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

5*n * (5*n - 4) * (5*n - 3) * (5*n - 2) * (5*n - 1) * (74370640*n^10 - 1858544992*n^9 + 20514338580*n^8 - 131664623360*n^7 + 543993683880*n^6 - 1511450439156*n^5 + 2859423911223*n^4 - 3636667338522*n^3 + 2975664813173*n^2 - 1414592080122*n + 296723700216) * a(n) = 4 * (602253442720*n^15 -16556130952016*n^14 + 205380792537480*n^13 - 1522719425465860*n^12 + 7532964897256240*n^11 - 26289347132202092*n^10 + 66724384353964514*n^9 - 125127125305800691*n^8 + 174348883354187484*n^7 - 179971361078129918*n^6 + 136029143462670914*n^5 - 73618072572837499*n^4 + 27462870106556136*n^3 - 6615958311698004*n^2 + 909677238744672*n - 52756794780480) * a(n-1) - 32 * (n - 1) * (268775492960*n^14 - 7254332587008*n^13 + 88047045148504*n^12 - 635783085242392*n^11 + 3044813388579412*n^10 - 10203895561701424*n^9 + 24599914682517462*n^8 - 43177965420907234*n^7 + 55190395890434461*n^6 - 50833857508826016*n^5 + 32977317197334769*n^4 - 14480415622830894*n^3 + 4020202694746872*n^2 - 620756474611752*n + 39040669694880) * a(n-2) + 128 * (n - 2) * (n - 1) * (93707006400*n^13 - 2388620193120*n^12 + 27114202589680*n^11 - 180957263558264*n^10 + 789422448157344*n^9 - 2367135714300940*n^8 + 4993922971646964*n^7 - 7459647015235870*n^6 + 7832732475984980*n^5 - 5661485242871255*n^4 + 2710593620426856*n^3 - 803964362776821*n^2 + 130003137365526*n - 8367104760840) * a(n-3) - 1024 * (n - 3) * (n - 2) * (n - 1) * (2751713680*n^12 - 63262737344*n^11 + 638112225172*n^10 - 3719043382864*n^9 + 13883533704384*n^8 - 34782677336708*n^7 + 59585768181167*n^6 - 69808362670470*n^5 + 55040889742427*n^4 - 28206394212700*n^3 + 8812932151470*n^2 - 1480802935914*n + 97764328620) * a(n-4) + 4096 * (n - 4) * (n - 3) * (n - 2) * (n - 1) * (2*n - 9) * (74370640*n^10 - 1114838592*n^9 + 7134112452*n^8 - 25533057632*n^7 + 56242855672*n^6 - 79077733668*n^5 + 71260022051*n^4 - 40254567238*n^3 + 13490580353*n^2 - 2379770358*n + 161791560) * a(n-5) for n > 4.

a(n) = [x^n] ( (1+x)^2 * (1+x^4) )^n.

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(n, k)*binomial(2*n, n+4*k));

Crossrefs

Cf. A259613, A395612, A395613.

Cf. A005809, A370196.

Sequence in context: A150158 A034010 A135588 * A318402 A293492 A150159

Adjacent sequences: A395611 A395612 A395613 * A395615 A395617 A395618

Keyword

nonn

Author

Seiichi Manyama, May 01 2026

Status

approved