A365730 G.f. A(x) satisfies A(x) = 1 + x^4*A(x)^4*(1 + x*A(x)).

1, 0, 0, 0, 1, 1, 0, 0, 4, 9, 5, 0, 22, 78, 91, 35, 140, 680, 1224, 969, 1254, 5985, 14630, 17710, 17710, 55660, 164450, 269100, 299520, 593775, 1805076, 3681405, 4951692, 7594752, 20173560, 47303520, 76404460, 110676324, 239784864, 589602585, 1106339923

Offset

0,9

Links

Robert Israel, Table of n, a(n) for n = 0..3345

Formula

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

From Seiichi Manyama, Sep 19 2025: (Start)

a(n) = (1/(n+1)) * [x^n] (1+x^4+x^5)^(n+1).

G.f.: (1/x) * Series_Reversion( x / (1+x^4+x^5) ). (End)

D-finite with recurrence: (6345612945*n^4 + 63456129450*n^3 + 222096453075*n^2 + 317280647250*n + 152294710680)*a(n) + (6485875002*n^4 + 80865731280*n^3 + 371068456410*n^2 + 740475262860*n + 539828550288)*a(n + 1) + (2903386045*n^4 + 54082374150*n^3 + 366487327415*n^2 + 1072456006230*n + 1144031904360)*a(n + 2) + (1595490360*n^4 + 35039295040*n^3 + 284053133520*n^2 + 1007484971360*n + 1319159099520)*a(n + 3) + (160749682*n^4 + 4678886244*n^3 + 48200943278*n^2 + 212653078476*n + 342679376280)*a(n + 4) + (-496345444*n^4 - 8868373024*n^3 - 58480367804*n^2 - 170808701024*n - 190903956960)*a(n + 5) + (-146022670*n^4 - 3313043700*n^3 - 27652488050*n^2 - 100392300540*n - 133453584360)*a(n + 6) + (-129258440*n^4 - 3564919360*n^3 - 36388614160*n^2 - 162501904760*n - 266908329120)*a(n + 7) + (-24008195*n^4 - 779791790*n^3 - 9452190625*n^2 - 50663185630*n - 101287463760)*a(n + 8) + (1430202*n^4 + 53651664*n^3 + 752447058*n^2 + 4674561876*n + 10850510160)*a(n + 9) + (53841*n^4 + 2288526*n^3 + 36464091*n^2 + 258131046*n + 685026720)*a(n + 10) + (768*n^4 + 36480*n^3 + 649680*n^2 + 5141400*n + 15255072)*a(n + 11) = 0. - Robert Israel, Mar 02 2026

Maple

f:= gfun:-rectoproc({(6345612945*n^4 + 63456129450*n^3 + 222096453075*n^2 + 317280647250*n + 152294710680)*a(n) + (6485875002*n^4 + 80865731280*n^3 + 371068456410*n^2 + 740475262860*n + 539828550288)*a(n + 1) + (2903386045*n^4 + 54082374150*n^3 + 366487327415*n^2 + 1072456006230*n + 1144031904360)*a(n + 2) + (1595490360*n^4 + 35039295040*n^3 + 284053133520*n^2 + 1007484971360*n + 1319159099520)*a(n + 3) + (160749682*n^4 + 4678886244*n^3 + 48200943278*n^2 + 212653078476*n + 342679376280)*a(n + 4) + (-496345444*n^4 - 8868373024*n^3 - 58480367804*n^2 - 170808701024*n - 190903956960)*a(n + 5) + (-146022670*n^4 - 3313043700*n^3 - 27652488050*n^2 - 100392300540*n - 133453584360)*a(n + 6) + (-129258440*n^4 - 3564919360*n^3 - 36388614160*n^2 - 162501904760*n - 266908329120)*a(n + 7) + (-24008195*n^4 - 779791790*n^3 - 9452190625*n^2 - 50663185630*n - 101287463760)*a(n + 8) + (1430202*n^4 + 53651664*n^3 + 752447058*n^2 + 4674561876*n + 10850510160)*a(n + 9) + (53841*n^4 + 2288526*n^3 + 36464091*n^2 + 258131046*n + 685026720)*a(n + 10) + (768*n^4 + 36480*n^3 + 649680*n^2 + 5141400*n + 15255072)*a(n + 11), a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1, a(5) = 1, a(6) = 0, a(7) = 0, a(8) = 4, a(9) = 9, a(10) = 5}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 02 2026

Prog

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

Crossrefs

Cf. A365727, A365728, A365729, A365731.

Cf. A001005, A001006, A365724.

Cf. A215341.

Sequence in context: A087369 A200629 A021206 * A121305 A002904 A200396

Adjacent sequences: A365727 A365728 A365729 * A365731 A365732 A365733

Keyword

nonn

Author

Seiichi Manyama, Sep 17 2023

Status

approved