A381840 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 - x^2*A(x)^7.

1, 1, 3, 11, 42, 153, 469, 690, -5967, -82708, -700876, -4989894, -32082336, -190742496, -1053280998, -5347579160, -24162468390, -88249158963, -157067396045, 1334548659436, 20996875910808, 194476989681546, 1491599102987040, 10232074769143770, 64440205192609155

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1233

Formula

a(n) = (1/(3*n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n+k,k) * binomial(4*n-k,n-2*k).

G.f.: ( (1/x) * Series_Reversion( x * (1-x+x^2)^3 ) )^(1/3).

D-finite with recurrence: (-69978919339*n^6 - 279915677356*n^5 - 449864481465*n^4 - 371316714860*n^3 - 165664380476*n^2 - 37772877744*n - 3426111360)*a(n) + (1251110982828*n^6 + 13647972998121*n^5 + 61470157316610*n^4 + 146112434704635*n^3 + 193204429674942*n^2 + 134737660010184*n + 38722219885440)*a(n + 1) + (-368290766853*n^6 - 4602181128003*n^5 - 23774771319345*n^4 - 64968490868085*n^3 - 99032752277082*n^2 - 79852579359192*n - 26625616859520)*a(n + 2) + (29823598494*n^6 + 419829506406*n^5 + 2421673685190*n^4 + 7328178903690*n^3 + 12284444512476*n^2 + 10845672563664*n + 3959445732480)*a(n + 3) + (-33657930*n^6 - 706816530*n^5 - 6151921650*n^4 - 28403553150*n^3 - 73363483620*n^2 - 100501747920*n - 57043958400)*a(n + 4) = 0. - Robert Israel, Mar 08 2026

Prog

(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n+k, k)*binomial(4*n-k, n-2*k))/(3*n+1);

Crossrefs

Cf. A103779, A367027.

Cf. A368971.

Sequence in context: A096147 A225431 A359248 * A099489 A077830 A393343

Adjacent sequences: A381837 A381838 A381839 * A381841 A381842 A381843

Keyword

sign

Author

Seiichi Manyama, Mar 08 2025

Status

approved